Poon Sandy SY 200911 MASc Thesis (1)

OPTIMIZATION OF SPAN-TO-DEPTH RATIOS IN HIGH-STRENGTH

CONCRETE GIRDER BRIDGES

by

Sandy Shuk-Yan Poon

A thesis submitted in conformity with the requirements

for the degree of Master of Applied Science

Graduate Department of Civil Engineering

University of Toronto

© Copyright by Sandy Shuk-Yan Poon (2009)

ii

Optimization of Span-to-Depth Ratios in High-Strength Concrete Girder Bridges

Sandy Shuk-Yan Poon

Master of Applied Science

Graduate Department of Civil Engineering

University of Toronto

2009

ABSTRACT

Span-to-depth ratio is an important bridge design parameter that affects structural behaviour,

construction costs and aesthetics. A study of 86 constant-depth girders indicates that conventional

ratios have not changed significantly since 1958. These conventional ratios are now questionable,

because recently developed high-strength concrete has enhanced mechanical properties that allow

for slenderer sections.

Based on material consumption, cost, and aesthetics comparisons, the thesis determines optimal

ratios of an 8-span highway viaduct constructed with high-strength concrete. Three bridge types are

investigated: cast-in-place on falsework box-girder and solid slabs, and precast segmental span-by-

span box-girder. Results demonstrate that total construction cost is relatively insensitive to span-to-

depth ratio over the following ranges of ratios: 10-35, 30-45, and 15-25 for the three bridge types

respectively. This finding leads to greater freedom for aesthetic expressions because, compared to

conventional values (i.e. 18-23, 22-39, and 16-19), higher ranges of ratios can now be selected

without significant cost premiums.

iii

ACKNOWLEDGEMENTS

I would like to express my deepest gratitude to my supervisor, Professor Paul Gauvreau, whose

encouragement, guidance, and support enabled me to complete this thesis.

I am also indebted to my research colleagues for their insightful advice and assistance

throughout my graduate studies: Cathy Chen, Billy Cheung, Davis Doan, Negar Elhami Khorasani,

Eileen Li, Kris Mermigas, Jason Salonga, Jimmy Susetyo, Brent Visscher, and Ivan Wu.

Lastly, I would like to thank my family for their support and encouragement over these past two

years.

iv

TABLE OF CONTENTS

Abstract .............................................................................................................................................. ii

Acknowledgements ........................................................................................................................... iii

Table of Contents ............................................................................................................................. iv

List of Figures ................................................................................................................................. viii

List of Tables .................................................................................................................................... xi

List of Symbols ............................................................................................................................... xiii

1 Introduction ............................................................................................................................... 1

1.1 The Significance of Optimizing Span-to-Depth Ratio ........................................................ 1

1.2 Objectives and Scope .......................................................................................................... 5

1.3 Thesis Structure ................................................................................................................... 6

2 Typical Span-to-Depth Ratios of Existing Bridges ................................................................. 7

2.1 Cast-in-Place Box-Girder .................................................................................................... 7

2.2 Cast-in-Place Slab ............................................................................................................. 12

2.3 Precast Segmental Box-Girder .......................................................................................... 16

2.4 Concluding Remarks ......................................................................................................... 18

3 Analysis Overview ................................................................................................................... 19

3.1 Analysis Model ................................................................................................................. 19

3.2 Materials ............................................................................................................................ 21

3.2.1 Prestressing Tendons ................................................................................................. 21

3.2.2 Concrete Covers ........................................................................................................ 22

3.3 Loads ................................................................................................................................. 22

3.3.1 Load Combinations and Load Factors ...................................................................... 22

3.3.2 Live Loads ................................................................................................................. 23

3.4 Design Requirements ........................................................................................................ 24

3.4.1 Ultimate Limit States Design Requirements ............................................................. 24

3.4.1.1 Flexural Strength ................................................................................................... 24

3.4.1.2 Shear Strength ....................................................................................................... 25

3.4.2 Serviceability Limit States Design Requirements ..................................................... 26

3.4.2.1 Stress ..................................................................................................................... 26

3.4.2.2 Vibration ............................................................................................................... 26

3.4.2.3 Deflection .............................................................................................................. 27

v

3.5 Other Preliminary Analysis Assumptions ......................................................................... 27

4 Analysis of Cast-in-Place on Falsework Bridges .................................................................. 28

4.1 Cast-in-Place on Falsework Construction ......................................................................... 28

4.2 Cast-in-Place on Falsework Box-Girder ........................................................................... 28

4.2.1 Model ........................................................................................................................ 28

4.2.1.1 Cross-Section ........................................................................................................ 29

4.2.1.2 Prestressing Tendon Layout .................................................................................. 30

4.2.2 Analysis Results ........................................................................................................ 31

4.2.2.1 Structural Behaviour and Dimensioning ............................................................... 31

4.2.2.2 Vibration Limits .................................................................................................... 32

4.2.2.3 Deflections ............................................................................................................ 33

4.2.2.4 Material Consumption ........................................................................................... 34

4.2.2.5 Limiting Factors of Span-to-Depth Ratios ............................................................ 36

4.3 Cast-in-Place on Falsework Solid Slab ............................................................................. 38

4.3.1 Model ........................................................................................................................ 39

4.3.1.1 Cross-Section ........................................................................................................ 39

4.3.1.2 Prestressing Tendon Layout .................................................................................. 39

4.3.2 Strip Method versus Beam Model ............................................................................. 40

4.3.3 Analysis Results ........................................................................................................ 42

4.3.3.1 Structural Behaviour and Dimensioning ............................................................... 42

4.3.3.2 Maximum Reinforcement Criterion ...................................................................... 43

4.3.3.3 Vibration Limits .................................................................................................... 44

4.3.3.4 Deflections ............................................................................................................ 45

4.3.3.5 Material Consumption ........................................................................................... 46

4.3.3.6 Limiting Factors of Span-to-Depth Ratios ............................................................ 47

5 Analysis of Precast Segmental Span-by-Span Box-Girder .................................................. 48

5.1 Precast Segmental Span-by-Span Construction ................................................................ 48

5.2 Model ................................................................................................................................ 49

5.2.1 Cross-Section ............................................................................................................ 50

5.2.2 Elevation and Prestressing Tendon Layout ............................................................... 50

5.3 Longitudinal Bending Moments ....................................................................................... 51

5.3.1 Construction Moments .............................................................................................. 51

5.3.2 Moments due to Thermal Gradient ........................................................................... 54

vi

5.4 Loss of Prestress ................................................................................................................ 57

5.4.1 Friction Losses .......................................................................................................... 57

5.4.2 Creep and Shrinkage Losses ..................................................................................... 58

5.4.3 Losses due to Relaxation of Prestressing Steel ......................................................... 59

5.4.4 Total Prestress Losses ............................................................................................... 59

5.5 Behaviour of Unbonded Tendons at Ultimate Limit States .............................................. 60

5.6 Analysis Results ................................................................................................................ 61

5.6.1 Structural Behaviour and Dimensioning ................................................................... 61

5.6.2 Vibration Limits ........................................................................................................ 61

5.6.3 Deflections ................................................................................................................ 62

5.6.4 Material Consumption ............................................................................................... 63

5.6.5 Limiting Factors of Span-to-Depth Ratios ................................................................ 64

6 Cost Comparisons ................................................................................................................... 65

6.1 Material Costs ................................................................................................................... 65

6.1.1 Material Unit Prices .................................................................................................. 65

6.1.1.1 Concrete Material Unit Price ................................................................................ 65

6.1.1.2 Cast-in-Place versus Precast Concrete .................................................................. 66

6.1.1.3 Falsework versus Erection Truss........................................................................... 67

6.1.1.4 Formwork .............................................................................................................. 67

6.1.1.5 Prestressing Tendons ............................................................................................. 67

6.1.2 Material Cost Comparisons ....................................................................................... 67

6.1.2.1 Concrete Cost Comparison ................................................................................... 68

6.1.2.2 Prestressing Cost Comparison ............................................................................... 69

6.1.2.3 Reinforcing Steel Cost Comparison ...................................................................... 70

6.1.2.4 Total Superstructure Cost ...................................................................................... 73

6.2 Overall Construction Costs ............................................................................................... 76

6.2.1 Construction Cost Breakdown .................................................................................. 76

6.2.2 Total Construction Cost Comparison ........................................................................ 77

6.3 Other Cost Factors ............................................................................................................ 78

6.4 Sensitivity Analysis ........................................................................................................... 79

6.4.1 Sensitivity with Respect to Changes in Material Unit Prices .................................... 79

6.4.2 Sensitivity with Respect to Changes in Construction Cost Breakdown .................... 83

6.5 Concluding Remarks ......................................................................................................... 85

vii

7 Aesthetics Comparisons .......................................................................................................... 86

7.1 Visual Impact of Span-to-Depth Ratio .............................................................................. 86

7.1.1 Effects of Viewing Points ......................................................................................... 92

7.1.2 Other Factors that Affect Visual Slenderness ........................................................... 94

7.2 Evolution of the Visually Optimal Span-to-Depth Ratio .................................................. 97

7.3 Concluding Remarks ....................................................................................................... 102

8 Conclusions ............................................................................................................................ 103

8.1 Conventional Span-to-Depth Ratios ............................................................................... 103

8.2 Maximum Span-to-Depth Ratios .................................................................................... 103

8.3 Material Consumption Comparisons ............................................................................... 104

8.4 Total Construction Cost Comparisons ............................................................................ 104

8.5 Aesthetic Comparisons.................................................................................................... 105

8.6 Optimal Span-to-Depth Ratios ........................................................................................ 105

Reference........................................................................................................................................ 107

Appendix A: Chapter 2 Supplementary Information ................................................................ 111

A.1 Cast-in-Place on Falsework Box-Girder ............................................................................ 112

A.2 Cast-in-Place on Falsework Solid Slab .............................................................................. 116

A.3 Precast Segmental Span-by-Span Box-Girder.................................................................... 118

Appendix B: Supporting Calculations ........................................................................................ 119

B.1 Flexural Strength for Bonded Tendons at ULS .................................................................. 119

B.2 Shear Strength at ULS ....................................................................................................... 120

B.3 Thermal Gradient Moments................................................................................................ 122

B.4 External Tendon Force ....................................................................................................... 124

B.5 Total Construction Cost ...................................................................................................... 125

Appendix C: Summary of Results ............................................................................................... 126

C.1 Cast-in-Place on Falsework Box-Girder ............................................................................. 127

C.2 Cast-in-Place on Falsework Solid Slab .............................................................................. 128

C.3 Precast Segmental Span-by-Span Box-Girder .................................................................... 129

C.4 Sensitivity with Respect to Changes in Construction Cost Breakdown ............................. 130

viii

LIST OF FIGURES

Figure 1-1. Recommended ratios for cast-in-place box-girder ......................................................... 2

Figure 1-2. Recommended ratios for cast-in-place slab .................................................................... 2

Figure 1-3. Recommended ratios for precast segmental box-girder ................................................. 2

Figure 2-1. Span-to-depth ratios of cast-in-place box-girders ........................................................ 10

Figure 2-2. Span-to-depth ratios of cast-in-place box-girders ........................................................ 11

Figure 2-3. Span-to-depth ratios of cast-in-place slabs ................................................................... 13

Figure 2-4. Span-to-depth ratios of cast-in-place slabs ................................................................... 14

Figure 2-5. Span-to-depth ratios of precast segmental box-girders ................................................ 17

Figure 2-6. Span-to-depth ratios of precast segmental box-girders ................................................ 17

Figure 2-7. Span-to-depth ratios for all bridge types ...................................................................... 18

Figure 3-1. Typical plan and elevation ........................................................................................... 19

Figure 3-2. Typical deck arrangement ............................................................................................ 19

Figure 3-3. Summary of analysis cases ........................................................................................... 20

Figure 3-4. Live loads: CL-625 truck load (top); CL-625 lane load (bottom) ................................ 23

Figure 3-5. Flexural resistance: a) cross-section, b) concrete stains, c) equivalent concrete stresses,

d) concrete forces .............................................................................................................................. 24

Figure 3-6. Construction cost economy from increasing the number of stirrup spacing ................ 25

Figure 3-7. Deflection limits for highway bridge superstructure vibration (CHBDC 2006) .......... 26

Figure 4-1. Moment comparison of bridges with constant and reduced end span length ............... 29

Figure 4-2. Typical cross-section for cast-in-place on falsework box-girder ................................. 29

Figure 4-3. Typical reinforcing steel layout .................................................................................... 30

Figure 4-4. Typical tendon profile .................................................................................................. 30

Figure 4-5. Changes in sectional modulus and cross-sectional depth ............................................. 32

Figure 4-6. Deflection for superstructure vibration limitation ........................................................ 33

Figure 4-7. Deflections: a) dead load, b) long-term, c) short-term ................................................. 33

Figure 4-8. Material consumptions for cast-in-place on falsework box-girder ............................... 35

Figure 4-9. Tendon arrangement that limits further increase in span-to-depth ratio ....................... 36

Figure 4-10. Interior box cavity limitation ...................................................................................... 37

Figure 4-11. Height of access diminishes as span-to-depth ratio increases .................................... 37

Figure 4-12. Concrete reduction due to increase in L/h ratio for solid slab and box-girder ........... 38

Figure 4-13. Voided slab ................................................................................................................. 38

Figure 4-14. Typical cross-section for cast-in-place on falsework solid slab ................................. 39

ix

Figure 4-15. Typical reinforcing steel layout .................................................................................. 39

Figure 4-16. Transverse distribution of longitudinal bending moment in slabs.............................. 41

Figure 4-17. Maximum reinforcement criterion: a) concrete stains, b) equivalent concrete stresses,

c) concrete forces .............................................................................................................................. 44

Figure 4-18. Deflection for superstructure vibration limitation ...................................................... 44

Figure 4-19. Deflections: a) dead load, b) long-term, c) short-term ............................................... 45

Figure 4-20. Material consumptions for cast-in-place on falsework solid slab .............................. 47

Figure 5-1. Precast segmental span-by-span construction method ................................................. 49

Figure 5-2. Span-by-span erection girder: a) overhead truss, b) underslung girder ........................ 49

Figure 5-3. Typical cross-section for precast segmental span-by-span box-girder ......................... 50

Figure 5-4. Typical reinforcing steel layout .................................................................................... 50

Figure 5-5. Typical tendon profile .................................................................................................. 51

Figure 5-6. Construction moments for segmental span-by-span method ........................................ 52

Figure 5-7. Redistribution of dead load moments due to creep ...................................................... 54

Figure 5-8. Redistribution of dead load and prestress moments due to creep ................................. 54

Figure 5-9. Thermal gradient effects ............................................................................................... 55

Figure 5-10. Moments due to thermal gradient ............................................................................... 56

Figure 5-11. Intentional angle changes ........................................................................................... 57

Figure 5-12. Long-term loss of prestress due to relaxation (Menn 1990) ....................................... 59

Figure 5-13. Compatibility conditions for bonded and unbonded tendons ..................................... 60

Figure 5-14. Deflection for superstructure vibration limitation ...................................................... 62

Figure 5-15. Deflections: a) dead load, b) long-term, c) short-term ............................................... 62

Figure 5-16. Material consumptions for precast span-by-span box-girder ..................................... 63

Figure 5-17. Access limited by height of interior box cavity .......................................................... 64

Figure 5-18. Access limited by height of interior box cavity and external tendons ........................ 64

Figure 6-1. Concrete material unit price ......................................................................................... 66

Figure 6-2. Concrete material cost comparison .............................................................................. 68

Figure 6-3. Total concrete cost comparison .................................................................................... 68

Figure 6-4. Prestressing tendon cost comparison ............................................................................ 69

Figure 6-5. Cost comparison of stirrups and minimum reinforcing steel ....................................... 71

Figure 6-6. Cost distribution of stirrups and minimum reinforcing steel ........................................ 72

Figure 6-7. Total reinforcing steel cost comparison ....................................................................... 73

Figure 6-8. Total superstructure material cost comparison ............................................................. 74

Figure 6-9. Total superstructure cost comparison (including cost of concrete placement)............. 75

x

Figure 6-10. Total construction cost comparison ............................................................................ 78

Figure 6-11. Total construction cost comparison (+50% concrete unit price) ................................ 80

Figure 6-12. Total construction cost comparison (-50% concrete unit price) ................................. 80

Figure 6-13. Total construction cost comparison (+50% prestressing tendon unit price)............... 81

Figure 6-14. Total construction cost comparison (-50% prestressing tendon unit price) ............... 81

Figure 6-15. Total construction cost comparison (+50% reinforcing steel unit price) ................... 82

Figure 6-16. Total construction cost comparison (-50% reinforcing steel unit price) .................... 82

Figure 6-17. Total construction costs under changes in construction cost breakdown ................... 84

Figure 7-1. Cast-in-place on falsework box-girder with L=50m .................................................... 87

Figure 7-2. Cast-in-place on falsework solid slab with L=30m ...................................................... 88

Figure 7-3. Precast segmental span-by-span box-girder with L=50m ............................................ 89

Figure 7-4. Visual effects of increasing span-to-depth ratios from 10 to 35 ................................... 90

Figure 7-5. Effect of increasing span length (box-girder with h=2.5m) ......................................... 91

Figure 7-6. Viewed from 300m ....................................................................................................... 92

Figure 7-7. Viewed from 150m ....................................................................................................... 92

Figure 7-8. Viewed from 75m ......................................................................................................... 92

Figure 7-9. Effects of pier width-to-height ratio and span-to-depth ratio ....................................... 93

Figure 7-10. Effect of span-to-depth ratio as viewing angle becomes less oblique ........................ 94

Figure 7-11. Effect of bridge height on perceived superstructure slenderness ............................... 95

Figure 7-12. Effect of pier configuration on perceived superstructure slenderness ........................ 95

Figure 7-13. Effect of deck cantilever length on perceived superstructure slenderness ................. 96

Figure 7-14. Glenfinnan Viaduct, 1901 (Cortright 1997) ............................................................... 97

Figure 7-15. Slender bridges by Maillart ........................................................................................ 98

Figure 7-16. Waterloo Bridge over the Thames (Darger 2002) ...................................................... 99

Figure 7-17. Changis-sur-Marne Bridge, 1948 (Mossot 2007) ....................................................... 99

Figure 7-18. Sketches to evaluate aesthetic impact of span-to-depth ratios (O'Connor 1991) ....... 99

Figure 7-19. Neckar Valley Viaduct, 1977 (Leonhardt 1982) ...................................................... 100

Figure 7-20. Kocher Valley Viaduct, 1979 (Leonhardt 1982) ...................................................... 100

Figure 7-21. Pregorda Bridge, 1974 (Menn) ................................................................................. 101

Figure 7-22. Felsenau Bridge, 1974 (Menn) ................................................................................. 101

Figure C-1. Cast-in-place on falsework box-girder with L=50m .................................................. 130

Figure C-2. Cast-in-place on falsework solid slab with L=25m ................................................... 130

Figure C-3. Precast segmental span-by-span box-girder with L=40m .......................................... 130

xi

LIST OF TABLES

Table 1-1. Description of recommended ratios ................................................................................. 3

Table 2-1. Summary of cast-in-place box-girders ............................................................................. 7

Table 2-2. Summary of cast-in-place slabs (continued) .................................................................. 13

Table 2-3. Summary of precast segmental box-girders ................................................................... 16

Table 3-1. Material properties ......................................................................................................... 21

Table 3-2. Material resistance factors (CSA 2006) ......................................................................... 21

Table 3-3. Prestressing tendon properties (CSA 1982) ................................................................... 21

Table 3-4. Corrugated metal duct properties (DSI 2008) ................................................................ 21

Table 3-5. Concrete cover requirements (CSA 2006) ..................................................................... 22

Table 3-6. Load combination .......................................................................................................... 22

Table 3-7. Load factors ................................................................................................................... 22

Table 3-8. DLA factor (CSA 2006) ................................................................................................ 23

Table 3-9. Multi-lane loading modification factor (CSA 2006) ..................................................... 23

Table 4-1. Summary of structural response and dimensioning of cast-in-place on falsework box-

girder ................................................................................................................................................. 31

Table 4-2. Summary of material consumptions for cast-in-place on falsework box-girder ............ 34

Table 4-3. Results from beam model and strip method .................................................................. 42

Table 4-4. Summary of structural response and dimensioning of cast-in-place on falsework solid

slab .................................................................................................................................................... 43

Table 4-5. Concrete strengths required to satisfy maximum reinforcement criterion ..................... 44

Table 4-6. Summary of material consumption for cast-in-place on falsework solid slab ............... 46

Table 5-1. Prestress losses due to friction ....................................................................................... 58

Table 5-2. Prestress losses due to anchorage set ............................................................................ 58

Table 5-3. Prestress losses due to creep and shrinkage ................................................................... 59

Table 5-4. Effective prestress after all losses .................................................................................. 59

Table 5-5. Prestress at ULS ............................................................................................................. 61

Table 5-6. Summary of structural response of precast span-by-span box-girder ............................ 61

Table 5-7. Summary of material consumption for precast span-by-span box-girder ...................... 63

Table 6-1. Material unit prices ........................................................................................................ 65

Table 6-2. Concrete material unit price ........................................................................................... 66

Table 6-3. Comparison of changes in cross-sectional depth and prestressing demand................... 70

Table 6-4. Total superstructure cost variations ............................................................................... 76

xii

Table 6-5. Construction cost breakdown (Menn 1990)................................................................... 77

Table 6-6. Material unit price changes ............................................................................................ 79

Table 6-7. Summary of material unit price sensitivity analysis ...................................................... 83

Table 6-8. Summary of cost study .................................................................................................. 85

Table C-1. Summary of results of cast-in-place on falsework box-girder analysis ...................... 127

Table C-2. Summary of results of cast-in-place on falsework solid slab analysis ........................ 128

Table C-3. Summary of results of precast segmental span-by-span box-girder analysis .............. 129

xiii

LIST OF SYMBOLS

A Gross cross-sectional area

Ac Area of concrete

Ap Area of prestressing steel

As Area of reinforcing steel

Av Stirrup area

C Compressive force

c Depth of compression region

e(x) Eccentricity of tendon at location x

Ec Concrete elastic modulus

Ep Prestressing tendon elastic modulus

Es Reinforcing steel elastic modulus

f’c Concrete compressive strength

fcr Concrete tensile strength

fpu Prestressing tendon ultimate strength

fpy Prestressing tendon yield stress

fr Free stress due to temperature gradient

fy Reinforcing steel yield stress

h Girder depth

I Moment of inertia

Ic Moment of inertia of gross uncracked concrete section

L Span length

L/h Span-to-depth ratio

lp Arc length of tendon between anchors

mop Moment when qp is applied to prestressing band

mp Moment when qp is applied to slab

Mr Flexural resistance

Mr Restraint moment

MSLS SLS moment demand

msp Self-equilibrating moment in strip method

MULS ULS moment demand

n Distance from base of cross-section to neutral axis (Section 5.3.2); Ep/Ec (Section

5.4.2)

xiv

P Prestressing force

P0 Jacking force

Pr Axial restraint force

qp Prestressing deviation force

S Sectional modulus

s Stirrup spacing

T Tensile force

z Moment lever arm

α(x) Sum of angle changes of tendon between stressing locations and point x

αc Thermal coefficient of concrete

αD Dead load factor

αp Prestress load factor

αx Intentional angle change of tendon

∆ Deflection

∆P Loss of prestress force

∆α Unintentional angle change of tendon

∆σp,rel Prestress loss due to relaxation of steel

ε0 Final strain

εc Concrete strain

εcs (t) Time-varying shrinkage strain

εcu Ultimate strain for concrete

εf Free strain due to temperature gradient

θ(y) Thermal differential

μ Coefficient of friction

σp0 Jacking stress

σp∞ Effective prestress after all losses

φ(t) Creep coefficient

φc Concrete resistance factor

φp Prestressing tendon resistance factor

φs Reinforcing steel resistance factor

ψ Final curvature of bending

1

1 INTRODUCTION

1.1 The Significance of Optimizing Span-to-Depth Ratio

Span-to-depth ratio, also known as slenderness ratio (L/h), is an important bridge design

parameter that relates a bridge’s span length to its girder depth. In the industry, this ratio is usually

used to establish the superstructure depth and is chosen during the conceptual design phase before

detailed calculations are performed. Selecting the ratio at an early stage of the design process

permits approximate dimensional proportioning which is needed for preliminary analysis to

evaluate the feasibility, cost-efficiency, and aesthetic merits of the design in comparison with

alternative design concepts (ACI-ASCE 1988). The ratio is commonly chosen based on experience

and typical values used in previously constructed bridges with satisfactory performance in order to

ensure that the design does not deviate drastically from past successful practice. The ratio can also

be determined by optimizing the combination of span length and superstructure depth to create a

cost-efficient and aesthetically-pleasing structure, but this generally involves an iterative process.

Therefore, instead of optimizing the span-to-depth ratio for every design concept, it is more

common to select ratios from a range of conventional values.

The choice of slenderness ratio is particularly critical in the design of girder-type bridges,

because it directly affects the cost of materials and construction of the superstructure. For instance,

using a high ratio (i.e. slender girder) reduces the concrete volume, increases the prestressing

requirement, and simplifies the construction due to a lighter superstructure. Moreover, slenderness

ratio has significant aesthetic impact, because the overall appearance of a girder-type bridge is

highly dependent on the proportion of the superstructure (Leonhardt 1982).

As stated previously, despite the significance of span-to-depth ratio, the industry has generally

relied on the same proven range of ratios over the past decades. Figures 1-1 to1-3 show the

recommended ranges of slenderness ratios outlined in different publications for three types of

prestressed concrete constant-depth girders: cast-in-place box-girder, cast-in-place slab, and precast

segmental box-girder. A brief description of the recommendations from each publication is given in

Table 1-1.

2

Figure 1-1. Recommended ratios for cast-in-place box-girder

Figure 1-2. Recommended ratios for cast-in-place slab

Figure 1-3. Recommended ratios for precast segmental box-girder

Menn 1990

Leonhardt 1979 Cohn & Lounis 1994

ACI-ASCE 1988

Duan et al. 1999AASHTO 1994

Leonhardt 1979Hewson 2003

0

10

20

30

40

1975 1980 1985 1990 1995 2000 2005 2010

Span-to-depth ratio

Year

Multiple-cell box-girder

Incremental launching method

Hewson 2003Cohn & Lounis 1994

Menn 1990

AASHTO 1994

Cohn & Lounis 1994

Leonhardt 1979

ACI-ASCE 1988

0

10

20

30

40

50

1975 1980 1985 1990 1995 2000 2005 2010

Span-to-depthratio

Year

Voided slab

Solid slab

Gauvreau 2006

ACI-ASCE 1988

AASHTO-PCI-ASBI 1997

Duan et al. 1999

0

5

10

15

20

25

1975 1980 1985 1990 1995 2000 2005 2010

Span-to-depthratio

Year

3

Table 1-1. Description of recommended ratios

Author Year Description

Leonhardt 1979 Fritz Leonhardt, a Professor of Civil Engineering at the University of Stuttgart, suggests ratios based on values from previously constructed prestressed concrete bridges with good performance. For cast-in-place single-cell box-girder, a ratio of 21 is recommended. The suggested ratio is lowered to around 12 to 16 when incremental launching method is used due to the large negative construction moments associated with this construction method. For cast-in-place slab, he suggests values from 18 to 36, with the higher values used for longer spans and for bridges with lighter traffic.

ACI-ASCE 1988 The American Concrete Institute-American Society of Civil Engineers (ACI-ASCE) Committee 343 on Concrete Bridge Design defines span-to-depth ratio recommendations for common bridge types based on typical values. These recommendations are intended to provide general guidelines for preliminary design. For cast-in-place, post-tensioned multiple-cell box-girder, ACI-ASCE recommends ratios from 25 to 33. The recommended ratio for precast multiple-cell continuous box-girder is around 22. These ratios are higher than the ones for single-cell box-girder, because a multiple-cell box section has more webs to accommodate tendons compared to a single-cell section with similar width. The recommended range of ratios is between 24 and 40 for cast-in-place, post-tensioned slab.

Menn 1990 Christian Menn is a Professor of Structural Engineering at the Institute of Structural Engineering in Zurich. His suggestions are based on existing bridges with satisfactory performance in terms of structural behaviour, aesthetics, and economics. He recommends ratios between 17 and 22 for cast-in-place box-girders, because girders with ratios below 17 would appear too heavy. On the other hand, girders with ratios above 22 have substantial cost increase due to the significantly higher longitudinal prestressing demand. Menn also suggests a maximum practical limit of 25 for solid slab and a maximum cost-effective slab depth of 0.8m.

AASHTO 1994 The American Association of State Highway and Transportation Officials (AASHTO) defines optional criteria for span-to-depth ratios in Cl.2.5.2.6.3 of the LRFD Bridge Design Specifications. These values are based on traditional maximum ratios of constant-depth continuous highway bridges with adequate vibration and deflection response. To ensure proper vibration and deflection behaviours, the maximum ratios are determined to be 25 for cast-in-place box-girder and 37 for cast-in-place slab.

Cohn & Lounis

1994 M.Z. Cohn is a Professor of Civil Engineering at the University of Waterloo and the span-to-depth ratios suggested in this paper are part of the results of a Ph.D. thesis prepared by Z. Lounis. These ratios are established from a systematic, multi-level optimization approach that determines the ideal cross-sectional dimensions, span layouts and superstructure system based on cost, material consumption, and aesthetics. For cast-in-place single-cell box-girder, the optimum ratio is found to range from 12 to 20. The ratio increases with span length and decreases with bridge width (e.g. a ratio of 12 corresponds to a span of 20m and a width of 16m while a ratio of 20 corresponds to a span of 50m and a width of 8m). This range of ratios is slightly lower relative to the ones from other publications, because this study investigates a simply-supported system while the ratios from other publications are mostly based on continuous systems. A simply-supported girder tends to be deeper since it experiences greater moments at midspan compared to a continuous structure. Cohn & Lounis also suggest the range of optimum ratios for voided and solid slabs are 22 to 29 and 28 to 33 respectively.

AASHTO-PCI-ASBI

1997 The American Segmental Bridge Institute (ASBI) has established various standard precast sections for segmental construction to enhance uniformity and simplicity for forming and production methods. Using these standard sections generally lead to practical and cost-effective solutions. The ranges of span-to-depth ratios obtained from these standard sections are 17 to 19 for span-by-span method and 17 to 20 for balanced cantilever method.

Duan et al. 1999 Lian Duan is a Senior Bridge Engineer with the California Department of Transportation and a Professor of Structural Engineering at Taiyuan University of Technology in China. A span-to-depth ratio of 25 is recommended for cast-in-place multiple-cell box-girder based on typical values from existing bridges. A range of ratios from 12.5 to 20 is recommended for precast segmental box-girder. This range is based on frequently used standard precast sections from Federal Highway Administration (FHWA).

4

Table 1-1. Description of recommended ratios (continued)

Author Year Description

Hewson 2003 Nigel Hewson is a recognized expert in the design and construction of prestressed bridges and is an Associate Lecturer at the University of Surrey on this subject. He suggested a span-to-depth ratio of 20 for cast-in-place single-cell box-girder and a maximum ratio of 20 for cast-in-place voided slab. Both of these recommendations are based on typical values.

Gauvreau 2006 A span-to-depth ratio of 17 is recommended for precast segmental span-by-span constructed box-girder. This value corresponds to the lower limit of span length used for this construction method (30m) and the minimum height requirement of a box section to provide sufficient access space within the box (1.8m). The recommended ratio is lower than the one for cast-in-place box-girder, because a larger depth is needed to compensate for the reduced tendon eccentricity due to the use of external unbonded tendons.

As shown in the previous graphs, there has been no significant increase in the recommended

span-to-depth ratio since 1979 despite the advancement in material strengths and construction

technologies. Recent developments have resulted in high-strength materials which theoretically

should lead to more slender structural components and longer span lengths. In particular, high-

strength concrete with compressive strength of 40 to 140 MPa has been achieved by lowering the

water-to-cement ratio and incorporating chemical admixtures (Kosmatka et al. 2002). Because of

their enhanced mechanical properties like higher ultimate strengths and modulus of elasticity, high-

strength concrete structures can resist the same level of loads using slenderer sections, resulting in

lightweight structures. The reduction in self-weight is especially critical in long-span bridges,

because the dead load consumes approximately 75% of the load-bearing capacity in long-span

bridges constructed with normal-strength concrete (TRB 1990). High-strength concrete lowers the

dead load contribution by using thinner sections and improves the load-bearing capacity by

increasing strength, thus slenderer bridges with longer spans can be attained.

High-strength concrete has been applied to various types of structures. For instance, concrete

with compressive strength of 60 MPa is commonly used for large bridges in Europe (Muller 1999)

while the building industry has been using concrete with strengths of over 100 MPa for years

(Hassanain 2002). However, most short- and medium-span bridges are being constructed with

concrete strengths of less than 50 MPa, because high-strength concrete is more expensive,

especially if the designer still uses the typical span-to-depth ratios as defined decades ago based on

normal-strength concrete (Hassanain 2002). For instance, the unit price of concrete rises by about

68% when the compressive strength changes from 30 MPa to 60 MPa (Dufferin Concrete 2009).

This indicates a substantial material cost increase if the same guidelines for superstructure

proportioning of normal-strength concrete bridges are applied to high-strength concrete bridges,

causing the application of high-strength concrete in bridges to be economically unfeasible.

Therefore, with the advent of high-strength materials, recommended span-to-depth ratios need to be

5

updated to match the improvement in material strength and stiffness and to provide an economic

incentive for the application of these materials in bridges.

1.2 Objectives and Scope

The purpose of this thesis is to determine the ideal range of span-to-depth ratios for post-

tensioned girder bridges constructed with current high-strength materials based on aesthetic

comparisons and optimization parameters such as material consumption and total construction cost.

The three bridge types considered in this study are cast-in-place on falsework box-girder and solid

slab, and precast segmental span-by-span box-girder. The objectives of this study are summarized

as follows:

Provide a study on the evolution of span-to-depth ratios in concrete girder bridges

constructed over the past 50 years and establish a range of conventional ratios.

Determine the amount of prestressing and the concrete strength needed to satisfy safety

and serviceability requirements as a function of span-to-depth ratio for the three types

of bridge considered.

Compare the material consumptions and total construction costs for bridges with

different slenderness ratios and determine the most cost-effective ratios.

Investigate the sensitivity of the construction cost results with respect to changes in

material unit cost and construction cost breakdown.

Examine the visual impact of different span-to-depth ratios and especially evaluate the

aesthetic influence of using the cost-effective ratios instead of conventional ones.

Update the recommendations for span-to-depth ratios based on economic and aesthetic

considerations.

The results of this research are expected either to confirm that the conventional ratios are already

optimal for new high-strength materials or to demonstrate that more slender sections can be attained.

The study focuses on the superstructure only while the prestressing and concrete strength

demands for the substructure are not explicitly accounted for. Also, only bridges with typical span

lengths are analyzed in this study: 35m to 75m for cast-in-place box-girder, 20m to 35m for cast-in-

place solid slab, and 30m to 50m for precast segmental box-girder.

6

1.3 Thesis Structure

The thesis is organized in eight chapters:

Chapter 1 provides the background and motivation of optimizing span-to-depth ratio.

Chapter 2 examines the span-to-depth ratios of existing bridges and discusses their changes over

the past 50 years. This information along with the span-to-depth ratio recommendations described

in Chapter 1 leads to values for conventional slenderness ratios. These conventional ratios serve as a

basis for cost and aesthetic comparisons in the later chapters.

Chapter 3 outlines the general analysis model and method used for all three bridge types. It also

provides a breakdown on all the analysis cases that need to be considered and discusses specific

design criteria that must be satisfied.

The specific analysis models and analysis results for cast-in-place on falsework box-girder and

solid slab, and precast segmental span-by-span box-girder are described in Chapter 4 and 5

respectively. Analysis results include structural responses, material consumptions, and factors that

limit further increase in slenderness ratio. The construction method and design issues unique to each

bridge type are also discussed.

Chapter 6 compares the material costs and total construction costs for bridges with varying

span-to-depth ratios for the three bridge types. Optimal ratios with the lowest costs are determined

and in particular, cost savings associated with using the optimal ratios instead of conventional ones

are examined. Furthermore, a sensitivity analysis is performed to demonstrate the effects of

changing unit costs and total construction cost breakdown on the analysis results.

Chapter 7 explores the aesthetic impacts of varying span-to-depth ratios and discusses the

public perception on visually optimal ratios.

Chapter 8 provides a conclusion for this study by summarizing the optimal span-to-depth ratios

for the three bridge types as well as their improvement over conventional ratios in terms of material

consumptions, construction costs, and aesthetics. These optimal ratios lead to updated span-to-depth

ratio recommendations for bridges constructed with current high-strength materials.

7

2 TYPICAL SPAN-TO-DEPTH RATIOS OF EXISTING BRIDGES

This chapter describes a study of 86 existing constant-depth girder bridges and presents a

compilation of their span-to-depth ratios. Specifically, the study determines the range of ratios

typically used in the industry and examines its variations over the past 50 years. Three bridge types

are considered: cast-in-place box-girder, cast-in-place slab, and precast box-girder. A majority of

these bridges has span-to-depth ratios within the suggested ranges discussed in Chapter 1, indicating

that a representative sample of bridges has been used.

2.1 Cast-in-Place Box-Girder

First, the study investigates 44 constant-depth cast-in-place box-girders. Table 2-1 provides the

basic information as well as a cross-sectional drawing for each bridge. Additional information,

including the span arrangement, girder dimensions, designer and references, is given in Appendix

A.1. Figure 2-1 shows the span-to-depth ratios with respect to the span lengths and compares these

ratios with the recommended values described in Section 1.1. Figure 2-2 plots the ratios with

respect to the completion years in order to illustrate the trend in slenderness ratio over time.

Table 2-1. Summary of cast-in-place box-girders

Bridge no.

Name Location Span-to-depth ratio

Construction method

Cross-section

1 Grenz Bridge at Basel Switzerland 17.7 N/A

2 Sart Canal-Bridge Belgium 12.0 Incremental launching

N/A

3 Weyermannshaus Bridge

Switzerland 18.9 N/A

4 Eastbound Walnut Viaduct

U.S.A. 23.0 CIP on falsework

5 & 6 Taiwan High Speed Rail (1) & (2)

Taiwan 11.4 Span-by-span

7 Pregorda Bridge Switzerland 22.2 Span-by-span on falsework

8 & 9 Almese Viaduct & Condove Viaduct

Italy 18.2 Balanced cantilever

10 Gravio Viaduct Italy 18.2 Balanced cantilever

Legend: N/A = no data CIP = cast-in-place

Split cross-section:

8

Table 2-1. Summary of cast-in-place box-girders (continued)

Bridge no.


Construction method

Cross-section

11 Borgone Viaduct Italy 18.2 Balanced cantilever

12 Quadinei Bridge Switzerland 20.0 Span-by-span on falsework

13 Altstetter Viaduct Switzerland 21.6 N/A

14 Reuss Bridge Switzerland 17.7 CIP on falsework

15 Cerchiara Viaduct Italy 18.5 Balanced cantilever

16 Castello Viaduct Italy 18.5 Balanced cantilever

17 Costacole Viaduct Italy 18.5 Balanced cantilever

18 Ferroviario Overpass at Bolzano

Italy 28.1 N/A

19 Krebsbachtal Bridge Germany 12.9 Incremental launching

20 Shatt Al Arab Bridge Iraq 12.8 Incremental launching

21 Ancona Viaduct Italy 20.7 Segmental

22 Felsenau Bridge (approaches)

Switzerland 16.0 Span-by-span on falsework

23 La Molletta Viaduct Italy 20.8 Segmental

24 Fosso Capaldo Viaduct

Italy 20.8 Segmental

25 Sihlhochstrasse Bridge

Switzerland 29.5 N/A

26 to 28

Grosotto Viaduct, Grosio Viaduct, Tiolo Viadut

Italy 20.0 Balanced cantilever

29 Denny Creek Viaduct U.S.A. 20.9 N/A

30 Woronora River Bridge

Australia 14.7 Incremental launching

N/A

9

Table 2-1. Summary of cast-in-place box-girders (continued)

Bridge no.


Construction method

Cross-section

31 Valentino Viaduct Italy 20.0 Balanced cantilever

32 Giaglione Viaduct Italy 20.0 Balanced cantilever

33 Venaus Viaduct Italy 20.0 Balanced cantilever

34 Passeggeri Viaduct Italy 20.0 Segmental

35 Brunetta Viaduct Italy 20.0 Segmental

36 Pietrastretta Viaduct Italy 20.0 Segmental

37 Deveys Viaduct Italy 20.0 Segmental

38 Gruyère Lake Viaduct Switzerland 15.1 Span-by-span

39 Interstate 895 Bridge over James River (approaches)

U.S.A. 21.3 Balanced cantilever

N/A

40 Lätten Bridge Switzerland 18.1 N/A

41 Savona Mollere Viaduct

Italy 22.5 Segmental

42 Ruina Viaduct Italy 19.3 N/A

43 Weinland Bridge Switzerland 22.6 Span-by-span on falsework

44 Kocher Valley Bridge Germany 21.2 Balanced cantilever

10

Figure 2-1. Span-to-depth ratios of cast-in-place box-girders

Figure 2-1 demonstrates that all 44 cast-in-place box-girders have span lengths between 35.4m

and 138m as well as span-to-depth ratios that range from 11.4 to 29.5. The frequency plot on the top

shows that 42 out of 44 bridges (95%) investigated have span lengths from 35m to 75m which is the

typical range for constant-depth box-girders as suggested by Hewson (2003). Above the frequency

plot are bridge numbers that relate each data point to its corresponding bridge in Table 2-1. The

frequency plot on the right shows a large concentration of bridges that have span-to-depth ratios

between 17.7 and 22.6. In fact, 33 out of 44 bridges (75%) have ratios within the range of values

recommended by Menn (17 to 22) and Hewson (20) which are based on existing bridges with

satisfactory performance, indicating that the study sample is representative of typical bridges.

Most bridges have ratios below 25 which is the traditional maximum value that ensures

adequate vibration and deflection responses in cast-in-place box-girders according to American

design standards (AASHTO 1994). Only 2 bridges (i.e. bridge no. 18 and 25), one with a multiple-

cell box-girder and the other with twin parallel box-girders, have ratios above 25. These two bridges,

however, are within the range of ratios recommended by ACI-ASCE (1988) for post-tensioned cast-

in-place multiple-cell box-girders (25 to 33). Higher ratios are expected for these types of cross-

sections, because additional webs can help accommodate the large amount of prestressing tendons

associated with slender girders without sacrificing the efficiency of the tendon layout (i.e. lowering

the tendon eccentricity by placing tendons in vertical layers within the webs). Also, the decrease in

spacing between webs causes considerable reduction in transverse bending for wide cross-sections,

thus lowering the transverse prestressing requirement.

2002

2000

1974

1992

1967

19721992

1974

1975

1975

1980

2001

1992

1978

19941984

19581971

Incremental launching

20 (Hewson)

0

5

10

15

20

25

30

35

20 40 60 80 100 120 140 160

Span-to-depthratio

Span length (m)

*Shaded region = Menn's range (17 to 22)

25 (Lian et al. , AASHTO, and minimum value of ACI-ASCE)

―

―

―

―

――

―

――――――

――――

―

――

―――

―

――――

―

―――――――

―

―

―

―

―

――

1 5 20 30 40 42 43 44Bridge No.

||||||||||||||||| ||||| || ||| |||||||||| || | | | |

11

Furthermore, 6 bridges have ratios of less than 15. Out of these 6 bridges, 4 are constructed

with incremental launching which generally requires a deeper cross-section to resist the large

negative moments during construction. They have ratios between 12 and 16 which is the typical

range for incrementally launched single-cell box-girders recommended by Leonhardt (1979). The

other two are railway bridges which also need a larger depth due to the greater live loads and more

stringent serviceability requirements. For instance, ACI-ASCE (1988) suggested a typical span-to-

depth ratio of approximately 16 for cast-in-place multiple-cell box-girders that carry railroads.

Figure 2-2 describes the variation in span-to-depth ratios over time. Only 37 out of the 44

bridges investigated are included in this graph due to the lack of data on completion year. These 37

bridges were completed between 1958 and 2002, and no significant variation in span-to-depth ratios

is observed within this time span. The slenderness ratios commonly used by the industry have not

increased over time despite the improvements in material strengths and advancements in

construction technologies. As stated previously, 75% of these bridges follow the same guidelines

recommended by Menn in 1986 and Hewson in 2003.

Figure 2-2. Span-to-depth ratios of cast-in-place box-girders

20 (Hewson)

Incremental launching

0

5

10

15

20

25

30

35

1955 1965 1975 1985 1995 2005

Span-to-depthratio

Year

*Shaded region = Menn's range (17 to 22)

25 (Lian et al. , AASHTO, and minimum value of ACI-ASCE)

―

―

―

―

――

―

――――

――

――――

―

――

―

―

―

―

――――

―

―――――――

―

―

―

―

―

――

12

2.2 Cast-in-Place Slab

In addition to cast-in-place box-girders, 28 cast-in-place constant-depth slab bridges are also

investigated. Table 2-2 provides basic information as well as a cross-sectional drawing for each

bridge and the detailed bridge information is given in Appendix A.2. Figure 2-3 and 2-4 relate the

span-to-depth ratios to the span lengths and to the completion dates.

Table 2-2. Summary of cast-in-place slabs

Bridge no.


Construction method

Cross-section

45 Khandeshwar Bridge India 20.3 N/A N/A

46 Spadina Ave. Bridge #16, Hwy 401

Canada 22.2 CIP on falsework

47 Spadina Ave. Bridge #18A, Hwy 401


48 Spadina Ave. Bridge #18B, Hwy 401




50 Spadina Ave. Bridge #21A, Hwy 401


51 Spadina Ave. Bridge #21B, Hwy 401


52 Sindelfingen Footbridge Germany 55.7 N/A

53 L 333 Overpass at Bassum Germany 21.9 N/A N/A

54 Waiblingen Footbridge Germany 42.5 N/A N/A

55 Mako Bridge Senegal 23.9 Incremental launching

N/A

56 Kittelbaches Bridge Germany 19.3 N/A N/A

57 San Francisco Airport Viaduct U.S.A. 19.2 N/A N/A

58 St. Vincent Street Overpass Canada 35.6 CIP on falsework

59 Bridge across Jan-Wellen-Platz Germany 25 N/A N/A







63 Saale Bridge at Rudolphstein Germany 20 N/A N/A

64 Bridge #20 at Hwy 401/427 Interchange




66 Spadina Ave. Bridge #11 Hwy 401

Canada 30 CIP on falsework

13

Table 2-2. Summary of cast-in-place slabs (continued)

Bridge no.


Construction method

Cross-section





69 Hundschipfen Bridge Switzerland 40.5 N/A N/A

70 McCowan Road Underpass Canada 37.9 CIP on falsework





Figure 2-3. Span-to-depth ratios of cast-in-place slabs

As shown in Figure 2-3, the 28 bridges have span lengths between 13.2m and 47.5m and span-

to-depth ratios from 19.2 to 55.7. The sample consists of 14 solid slabs and 14 voided slabs. Out of

the 14 solid slabs, 7 that were mostly built in the 1960s have span lengths greater than 20m which is

the current maximum economic span length for this type of slab (Gauvreau 2006). All voided slabs,

except for bridge no. 72, have spans of less than the maximum typical span length of 46m as

suggested by ACI-ASCE (1988). Also, most of the bridges (79%) have span-to-depth ratios that are

within Leonhardt’s recommended range of 18 to 36 and are below AASHTO’s maximum value for

45 52 56 58 60 63 68 70 72Bridge No.

1993

1996

1975

1992 19942000

1964

1963

1963

1963 1963

1967

1986

1961

1963

1963

2000

196337 (AASHTO)

25 (Menn, Cohn & Lounis for voided slab)

30.5 (Cohn & Lounis for solid slab)

20 (Hewson for voided slab)

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Span-to-depthratio

Span length (m)

Voided slab

Solid slab

*Shaded region = Leonhardt's range (18 to 36)

―――――――

―

―

―

―

――

―

―

―――

―

―

―――

―

――

| |||||| ||| ||| || ||| | | || | | | |

14

adequate deflection and vibration behaviour. Since the majority of the sample has span-to-depth

ratios within Leonhardt’s suggested range and spans similar to conventional values, the sample is

fairly representative of typical slab bridges.

Figure 2-4. Span-to-depth ratios of cast-in-place slabs

As shown in Figures 2-3 and 2-4, most of the bridges have span-to-depth ratios that cluster

around two ranges: 13 bridges (46%) have ratios between 19 and 25 while 13 bridges have ratios

between 30 and 40.5. The first range is composed of 7 solid slabs built in the 1960s and 6 voided

slabs built after 1970. The latter range consists of 6 solid slabs and 7 voided slabs which are all

constructed in the 1960s except for bridge no. 69. The remaining two bridges (bridge no. 52 and 54)

with higher ratios of 42.5 and 55.7 are both pedestrian bridges which can be more slender due to the

lower live load requirements. Therefore, there is a noticeable variation in the typical range of span-

to-depth ratios depending on the construction year and function of the bridge.

Figure 2-4 clearly illustrates the changes in typical span-to-depth ratios with respect to

construction year. Out of the 19 bridges completed prior to 1975, 12 (63%) have slenderness ratios

greater than 30 and 12 (63%) are solid slabs. Newer bridges are mainly voided slabs with lower

slenderness ratios at around 20 due to the stricter code requirements in recent years. For instance,

the Ontario Ministry of Transportation (MTO) sets the minimum non-prestressed reinforcement

clear cover to be 70±20 for the top surface of voided slabs in the MTO Structural Manual (2003)

while the value is only 50±20 in the Ontario Highway Bridge Design Code (OHBDC 1983).

Likewise, the MTO Structural Manual limits the maximum span-to-depth ratio to 28 for all post-

tensioned slabs while no such provisions existed prior to 1975 (Scollard and Bartlett 2004). As

shown in Figure 2-4, the first generation post-tensioned voided slabs constructed in the 1960s have

37 (AASHTO)

25 (Menn, Cohn & Lounis for voided slab)

30.5 (Cohn & Lounisfor solid slab)

20 (Hewson for voided slab)

0

10

20

30

40

50

60

1960 1970 1980 1990 2000 2010

Span-to-depthratio

Year

Voided slab

Solid slab

*Shaded region = Leonhardt's range (18 to 36)

―――――――

―

―

―

―

――

―

―

―――

―

―

―――

―

――

15

span-to-depth ratios of over 30 which required large amount of longitudinal prestressing. This

resulted in the formation of longitudinal cracks above the voids due to the large concentrated post-

tensioning forces near the abutments, which created transverse splitting stresses, and due to the

restraint of transverse concrete shrinkage imposed by the steel void forms. To solve this cracking

problem, MTO recommended the addition of transverse prestressing to prevent shrinkage cracking

and a decrease in span-to-depth ratio in 1975 in order to reduce the required prestressing force,

which eventually led to the current maximum span-to-depth ratio limit of 28 (Scollard and Barlett

2004). As a result, in this study, 6 out of the 9 bridges built after 1975 have span-to-depth ratios

below 28; the remaining 3 bridges are pedestrian bridges or European bridges.

As stated before, the typical span-to-depth ratios for slab bridges vary considerably with time

and bridge function. The impact of slab type (i.e. solid or voided), on the other hand, is not as

significant. According to the literature discussed in Chapter 1, the conventional span-to-depth ratios

for solid slab are expected to be higher than the ones for voided slab, because voided slabs are

commonly used to reduce self-weight for longer spans that require slabs thicker than 800mm (Menn

1990). In fact, the study by Cohn and Lounis (1994) suggested that the optimum depth for voided

slab is 12% to 20% thicker than the one for solid slab, resulting in optimum ratios of approximately

30.5 for solid slab and 25 for voided slab. The sample in this study indicates a small difference in

span-to-depth ratios between the two slab types. Voided slabs have ratios that range from 19 to 35

while the range is from 22 to 39 for solid slabs (excluding the pedestrian bridges). These results are

reasonable, because a voided slab is theoretically an intermediate cross-section between a solid slab

and a box-girder and its range of ratios is expected to be in between the ones from solid slab and

box-girder (i.e. 17 to 22 as determined in Section 2.1). The typical ratios of the voided slab might be

closer to those of the solid slab or of the box-girder depending on its component dimensions. For

instance, if the void diameter is less than 60% of the total slab depth, the longitudinal behaviour

would resemble a solid slab (O'Brien and Keogh 1999). In this sample, the mean ratio for solid slab

is 30 which is only slightly higher than the 27 for voided slab if all the bridges from 1960 to 2000

are considered. If only recently constructed bridges are considered (i.e. built after 1990), the

conventional ratio for voided slab would decrease to around 20 which is the same as Hewson’s

suggested value. There is no data for recently constructed solid slabs, but the conventional ratio for

solid slab is expected to follow the same trend when the entire sample is considered and be only

slightly higher than the value for voided slab.

16

2.3 Precast Segmental Box-Girder

In this section, 14 precast segmental box-girders are examined. Table 2-3 provides basic

information as well as a cross-sectional drawing for each bridge and the detailed bridge information

is given in Appendix A.3. Figures 2-5 and 2-6 relate the span-to-depth ratios to the span lengths and

the completion dates.

Table 2-3. Summary of precast segmental box-girders

Bridge no.


Construction method Cross-section

73 Bukit Panjang LRT System 801

Singapore 15.7 Segmental span-by-span

74 Wiscasset Bridge U.S.A. 17.5 Segmental with launching girder

N/A

75 Chiovano Viaduct Italy 17 Segmental

76 Collecastino Viaduct Italy 17 Segmental

77 Fiumetto Viaduct Italy 17 Segmental

78 San Leonardo Viaduct Italy 17 Segmental

79 Petto Viaduct Italy 17 Segmental

80 Cadramazzo Viaduct Italy 16.7 Balanced cantilever

81 Fella IX Viaduct Italy 16.7 Balanced cantilever

82 Malborghetto Viaduct Italy 16.1 Balanced cantilever

83 Val Freghizia Viaduct Italy 16.8 Balanced cantilever

84 Fella IV Viaduct Italy 17.6 Balanced cantilever

85 Ngong Shuen Chau Viaduct

China 18.8 Balanced cantilever

86 Sutong Bridge Approach (Nantong side)

China 18.8 Balanced cantilever

17

Figure 2-5. Span-to-depth ratios of precast segmental box-girders

Figure 2-6. Span-to-depth ratios of precast segmental box-girders

According to Figure 2-5, all 14 bridges have slenderness ratios between 15.7 and 18.8 which are

within the range of frequently used ratios suggested by Duan et al. (1999). Also, 13 bridges have

spans between 30m and 60m which is a feasible and cost-effective span range for precast segmental

constructed constant-depth girders recommended by ASBI (1997). Since most of the bridges in this

study sample have span-to-depth ratios and span lengths within standard ranges, these bridges are

assumed to be representative of typical precast segmental box-girders.

1998

1992

1986

1981

1986

1988 1985

20072007

17 (Gauvreau)

0

5

10

15

20

25

0 10 20 30 40 50 60 70 80 90

Span-to-depthratio

Span length (m)

*Shaded region = range from Duan et al. (12.5 to 20)

―

―――――――――――――

17 (Gauvreau)

0

5

10

15

20

25

1980 1985 1990 1995 2000 2005 2010

Span-to-depthratio

Year

*Shaded region = range from Duan et al. (12.5 to 20)

―

――――――――――――

| | ||||||| | | || |73 75 83 84 86Bridge No.

18

Moreover, out of the 14 bridges, 10 bridges (71%) have ratios within 5% from 17 which is the

recommended value for precast segmental span-by-span construction from Gauvreau (2006). The

lowest ratio is 15.7 for bridge no. 70 which is a railway bridge that requires a deeper girder to

satisfy the more stringent serviceability requirements. Also, Figure 2-6 indicates that the typical

span-to-depth ratios did not vary significantly from 1981 to 2007.

2.4 Concluding Remarks

This study examines the span-to-depth ratios of 86 constant-depth girder bridges in order to

determine the range of ratios typically used by the industry over the past 50 years. The slenderness

ratios with respect to span lengths for all of these bridges are illustrated in Figure 2-7. The average

span-to-depth ratio and the typical span lengths for each bridge type are also indicated on the graph.

Figure 2-7. Span-to-depth ratios for all bridge types

The primary findings of this investigation are summarized in Table 2-4. Average ratios within

the typical ranges are considered as the conventional ratios and are used as a basis of comparison in

this thesis.

Table 2-4. Summary of conventional span-to-depth ratios

Bridge type Range of span-to-depth ratios

Number of bridges within this range

Average ratio

Notes

Cast-in-place box-girder 17.7 to 22.6 33 out of 44 (75%) 20 Range varies little between 1958 and 2002

Cast-in-place voided slab 19 to 35 13 out of 14 (92%) 27 Conventional ratio is closer to 20 for bridges completed after 1990

Cast-in-place solid slab 22 to 39 12 out of 14 (86%) 30 Used mainly from 1961 to 1975

Precast segmental box-girder

15.7 to 18.8 14 out of 14 (100%) 17 Range varies little between 1981 and 2007

20 (CIP box-girder)

27 (CIP voided slab)

30 (CIP solid slab)

17 (precast segmental box-girder)

15 to 48m (CIP slab)

0

10

20

30

40

50

60

70

0 20 40 60 80 100 120 140

Span-to-depthratio

Span length (m)

CIP box-girder

CIP voided slab

CIP solid slab

Precast segmental box-girder

Average ratios:

30 to 60m (precast segmental box-girder)

Typical span lengths:

Average ratios:


Average ratios:


35 to 75m (CIP box-girder)

19

3 ANALYSIS OVERVIEW

This purpose of this analysis is to compute the amount of prestressing and the concrete strength

needed to satisfy design requirements for bridges with varying span lengths and span-to-depth ratios.

These material consumption results are then used to compute construction cost as a function of

span-to-depth ratio. By examining the variations in construction cost and aesthetic impacts, the

study determines the most cost-optimal ratios for different bridge types. The three post-tensioned

bridge types considered are cast-in-place on falsework box-girder, cast-in-place on falsework solid

slab, and precast segmental span-by-span box-girder.

This chapter describes the analysis model, material properties, applied loads, ultimate and

serviceability limit states design requirements, as well as some preliminary analysis assumptions.

The analysis is performed using the program SAP2000 and spreadsheet calculations.

3.1 Analysis Model

The analysis model is an 8-span highway viaduct with a straight profile. Typical plan and

elevation are shown in Figures 3-1.

Figure 3-1. Typical plan and elevation

Two types of constant-depth cross-sections are investigated: single-cell box-girder and solid

slab. Both types have 0.5m wide barriers, 90mm thick wearing surface and 3.5m wide design lanes

as shown in Figure 3-2. Proportions of other cross-sectional components vary for different

construction methods and are discussed in greater details in Sections 4.2.1.1, 4.3.1.1, and 5.2.1.

Figure 3-2. Typical deck arrangement

Design lane 1Design lane 2

Design lane 3Design lane 4

CL Bridge

Abutment AbutmentPier 1 Pier 2 Pier 3 Pier 4 Pier 5 Pier 6 Pier 7

Bridgewidth

b

PLAN

ELEVATION

Lend L L L L L L L

end

Depthh

Design Lane 1 Design Lane 2 Design Lane 3 Design Lane 4

90mm thick wearing surfaceCLBarrier

3500 3500 3500 3500 500500

20

In the analysis, the span length and span-to-depth ratio are varied in the model to generate the

analysis cases illustrated in Figure 3-3. It should be noted that for cast-in-place on falsework box-

girders, cases with spans of 75m are included in the study mainly for comparison purposes. In the

industry, however, such long spans are generally constructed with cantilever method in regions

where high labour costs deter the extensive use of falsework.

Legend: L = span length L/h = span-to-depth ratio

Figure 3-3. Summary of analysis cases

8-span highway viaduct

Cast-in-place on falsework box-girder

L = 35m

L/h = 10, 15, 20, 25

L = 50m

L/h = 10, 15, 20, 25, 30

L = 60m

L/h = 10, 15, 20, 25, 30, 35

L = 75m

L/h = 10, 15, 20, 25, 30, 35

Cast-in-place on falsework solid slab

L =20m

L/h = 30, 35, 40, 45

L = 25m

L/h = 30, 35, 40, 45

L = 30m

L/h = 30, 35, 40, 45, 50

L = 35m

L/h = 30, 35, 40, 45, 50

Precast segmental, span-by-span box girder

L = 30m

L/h = 15, 20, 25

L = 40m

L/h = 15, 20, 25, 30

L = 45m

L/h = 15, 20, 25, 30

21

3.2 Materials

Tables 3-1 and 3-2 summarize the material properties and resistance factors used in the analysis.

To illustrate the effects of high-strength materials on span-to-depth ratios, a concrete compressive

strength of 50 MPa is used in the analysis, because this value is the minimum strength requirement

of high performance concrete as defined by MTO (OPSS 2007).

Table 3-1. Material properties

1 For solid slab analysis, higher concrete strengths (i.e. up to 80 MPa) are used for slender cases. 2 For prestressing tendons and reinforcing steel bars, a bilinear stress-strain relationship is used:

fp = Epεp ≤ fpy

fs = Esεs ≤ fy

Table 3-2. Material resistance factors (CSA 2006)

Material Resistance factor φ

Concrete φc = 0.75 Prestressing strands φp = 0.95 Reinforcing steel bars and wires φs = 0.90

3.2.1 Prestressing Tendons

All analysis cases utilize size 15 seven-wire low-relaxation strands as prestressing tendons and

standard size corrugated metal ducts with properties summarized in Tables 3-3 and 3-4.

Table 3-3. Prestressing tendon properties (CSA 1982)

Property Value

Nominal diameter 15.24 mm Nominal Area 140 mm

2

Mass 1.109 kg/m

Table 3-4. Corrugated metal duct properties (DSI 2008)

# strands Outer diameter (mm) Minimum block-out diameter (mm) Transition length (mm)

12 94 254 508 15 97 279 575 19 106.3 305 640 27 121.4 343 702 37 138.4 407 890

Material Property Value

Concrete1

Compressive strength Tensile strength Elastic modulus

= 3000 fc′ + 6900

γc

2300

1.5

(CSA 2006)

f’c = 50 MPa

fcr = 2.8 MPa = 0.4 fc′ (CSA 2006)

Ec = 28100 MPa

Prestressing tendons2

Ultimate strength Yield stress Effective prestress after all losses Elastic modulus

fpu = 1860 MPa fpy = 0.9 fpu = 1670 MPa σp∞ = 0.6fpu = 1120 MPa Ep = 200000 MPa

Reinforcing steel bars2

Yield stress Elastic modulus

fy = 400 MPa Es = 200000MPa

22

The minimum clear distance between adjacent ducts is 40mm according to the Canadian

Highway Bridge Design Code (CHBDC) (CSA 2006). A horizontal spacing of about the duct

diameter is used between tendons to provide sufficient space for concrete placement and vibration.

3.2.2 Concrete Covers

Concrete cover requirements and tolerances from CHBDC are used (Table 3-5). The bridge is

assumed to be exposed to the most severe environmental category (i.e. de-icing chemicals, spray or

surface runoff containing de-icing chemicals, marine spray).

Table 3-5. Concrete cover requirements (CSA 2006)

Component Reinforcement or steel ducts Concrete covers (mm)

Cast-in-place Precast

Top surface of bottom slab Reinforcing steel 40 ± 10 40 ± 10 Post-tensioning ducts 60 ± 10 60 ± 10

Top surface of top slab Post-tensioning ducts 130 ± 15 120 ± 10

Soffit of top and bottom slabs Reinforcing steel 50 ± 10 45 ± 10 Post-tensioning steel 70 ± 10 65 ± 10

Vertical surfaces Reinforcing steel 70 ± 10 60 ± 10 Post-tensioning ducts 90 ± 10 80 ± 10

3.3 Loads

Dead loads, live loads (truck and lane), and prestress loads based on CHBDC are used in the

analysis. Thermal gradient effects are considered only in the precast segmental box-girder analysis

because they are more critical for precast girders which do not have continuous bonded steel.

3.3.1 Load Combinations and Load Factors

The load combinations and load factors used in the analysis under ultimate and serviceability

limit states are summarized in Table 3-6 and Table 3-7 (CSA 2006). Fatigue limit state is not

considered in the analysis, because the concrete is kept uncracked at service and the steel does not

experience large stress cycles.

Table 3-6. Load combination

Permanent loads Transitory loads Loads D P L K

Ultimate limit states ULS1 ULS2

αD

αD

αP

αP

1.70

0

0 0

Serviceability limit states SLS1 SLS2

SLS3

1.00 0

1.00

1.00 0

1.00

0.90 0.90

0

0.80 0 0

Table 3-7. Load factors

Dead load Load factor αD

Precast concrete 1.10 Cast-in-place concrete 1.20 Barriers 1.20 Wearing surfaces 1.50

Prestress load Load factor αP

Secondary prestress effects

0.95

Legend: D = dead load L = live load P = secondary prestress effects K = effects of strains due to temperature differential

23

As shown in Table 3-6, load combinations ULS2 and SLS3 only consider permanent loads.

Without live loads, longitudinal tendons push the bridge upward while there might not be enough

dead load to weigh the bridge down. Therefore, this load combination must be checked to ensure

ULS and SLS requirements are satisfied when the bridge is hogging. This is particularly critical for

slender bridges since they have large prestressing forces and small dead loads. On the other hand,

load combination SLS2 considers pure live loads when the bridge is loaded with only one CHBDC

CL-625 truck; this combination is used for superstructure vibration check.

3.3.2 Live Loads

The live loads considered in the analysis comprise of CL-625 truck and CL-625 lane loads as

illustrated in Figure 3-4.

Figure 3-4. Live loads: CL-625 truck load (top); CL-625 lane load (bottom)

The CL-625 truck load is increased by a dynamic load allowance (DLA) factor for SLS1 and

ULS load combinations as shown in Table 3-8 (CSA 2006, Cl. 3.8.4.5.3). This DLA factor accounts

for the load increase due to impact from truck vibrations and it depends on the number of axles that

are loaded to produce the maximum force effect. The more the axles, the lower is the DLA factor,

because the probability of all of the axles being in phase is low. Furthermore, the model is under

multi-lane loading and the traffic load moments are modified according to the number of design

lanes loaded to produce maximum force effects as shown in Table 3-9. These modification factors

account for the probability of simultaneously loading more than one lane.

Table 3-8. DLA factor (CSA 2006)

# of axles loaded DLA factor

1 1.40 2 1.30

3 or more 1.25

Table 3-9. Multi-lane loading modification factor (CSA 2006)

# of loaded design lanes Modification factor

1 1.00 2 0.90 3 0.80 4 0.70 5 0.60

6 or more 0.55

50 125 125 175 150Axle loads, kN

Axle no. 1 2 3 4 5

3.6 m 1.2 m 6.6 m 6.6 m

18 m

3.6 m 1.2 m 6.6 m 6.6 m

18 m

40 100 100 140 120Axle loads, kN

CL-625 Truck Load

Uniformly distributed load 9kN/m

CL-625 Lane Load

50 125 125 175 150Axle loads, kN

Axle no. 1 2 3 4 5

3.6 m 1.2 m 6.6 m 6.6 m

18 m

3.6 m 1.2 m 6.6 m 6.6 m

18 m

40 100 100 140 120Axle loads, kN

CL-625 Truck Load

Uniformly distributed load 9kN/m

CL-625 Lane Load

24

3.4 Design Requirements

All analysis cases are designed to satisfy CHBDC ULS and SLS requirements by increasing the

amount of prestressing tendons and stirrups. Also, for the solid slab analysis, concrete strengths are

increased (up to 80 MPa) in order to reduce the prestressing demands in slender cases since the

dimensions of such cases cannot accommodate the large amount of tendons required to satisfy ULS.

The ULS check consists of flexural and shear strength requirements while the SLS check includes

stress, vibration and deflection limitations. In the analysis, such design checks only consider

longitudinal behaviour and ignore transverse behaviour based on the assumption that transverse

reinforcement demands remain constant for analysis cases with the same span lengths and bridge

type. This assumption is valid, because transverse behaviour depends highly on live loads which are

the same for cases with the same span lengths. Although dead loads vary for these analysis cases,

they contribute very little to the transverse calculations due to the small influence lengths. This

assumption does not affect the results of this comparative study in which comparisons are made

between bridges with the same span length but different span-to-depth ratios, and the relative values

of material consumptions and construction costs are more important than the actual values.

3.4.1 Ultimate Limit States Design Requirements

This section describes the ultimate limit states design requirements which include flexural and

shear strength checks.

3.4.1.1 Flexural Strength

In ULS, the flexural resistance (Mr) is a pure couple between compression in the concrete and

tension in the longitudinal prestressing tendons. The analysis assumes that other non-prestressed

reinforcements do not contribute to flexural strength and that cracked concrete has no tensile

strength. Also, an equivalent rectangular concrete stress distribution is used as shown in Figure 3-5.

Figure 3-5. Flexural resistance: a) cross-section, b) concrete stains, c) equivalent concrete stresses, d) concrete forces

Therefore, Mr = T∙z = φpfpyApz for cases with bonded tendons in which the changes in tendon

strain are assumed to be equal to the changes in strain of the surrounding concrete. A sample

calculation is shown in Appendix B.1. For cases with unbonded tendons (i.e. span-by-span box-

girders), this assumption is no longer valid and a different approach is used (refer to Section 5.5).

c

εc= 3.5x10-3

φ a =β

1c

fpy

α1f

c’

T = φpA

pf

py

C

z Mr= Tz

Strain Stress Forces

25

Furthermore, since the analysis model is a statically indeterminate system, negative bending

moments exist around the supports. In such regions, the bottom slab thickness of the box-girder is

proportioned such that the compressive depth c is within the bottom flange.

Other flexural requirements that are considered include minimum and maximum reinforcement

requirements. The first requirement states that the flexural resistance must be greater than 1.2 times

the cracking moment (Mcr) or 1.33 times the moment demand at ULS, whichever value is smaller

(CSA 2006, Cl. 8.8.4.3). The second requirement states that c/d should be less than 0.5 (refer to

Figure 3-5) (CSA 2006, Cl. 8.8.4.5). This requirement ensures that the steel has yielded when the

concrete crushes at a strain of 0.0035 at the extreme compression fibre and that significant plastic

deformation has developed prior to failure.

3.4.1.2 Shear Strength

The CHBDC sectional design model is used to compute shear strengths and a sample shear

calculation is shown in Appendix B.2. Shear resistance (Vr) comprises of three components:

concrete, prestressing tendons, and stirrups. Concrete shear resistance depends on the cross-section

while the prestressing shear resistance depends on the vertical component of prestressing force

which is determined by flexural requirements. Therefore, the only independent variable that can

increase shear strength is the amount of stirrups. In the analysis, the stirrups are at least 20M and the

minimum spacing of stirrups is 300mm as required by CHBDC to support longitudinal tendon ducts

and this spacing is reduced if it does not provide sufficient shear resistance (CSA 2006, Cl. 8.14.6).

Each analysis case uses at most two different stirrup spacing since further discretization yields little

changes in the final results as shown in Figure 3-6.

Figure 3-6. Construction cost economy from increasing the number of stirrup spacing

Figure 3-6 compares the cost results of for an analysis case when the number of stirrup spacing

increases from one to four. The graph is obtained by computing the material consumption and total

construction cost for a span-by-span constructed, precast segmental box-girder with a span length of

40m and span-to-depth ratio of 20. As the number of stirrup spacing increases, the amount of

$277 $260 $258 $257

$2,290 $2,273 $2,271 $2,270

$0

$500

$1,000

$1,500

$2,000

$2,500

1 2 3 4

Cost per deck area

($/m2)

Number of stirrup spacing

Steel cost

Total construction cost

26

stirrups needed decreases, because the stirrup layout is further refined to match the shear demand.

This trend is illustrated in the graph, but the graph also indicates that the savings in steel diminishes

as the number of stirrup spacing increases beyond two. For instance, the steel cost decreases by 6%

when two spacing instead of one spacing are used. On the other hand, the steel cost decreases by

only 1% when the number of spacing increases from two to four. Therefore, only two different

stirrup spacing are used in the analysis.

3.4.2 Serviceability Limit States Design Requirements

This section describes the serviceability limit states design requirements which comprise of

stress, vibration and deflection limitations.

3.4.2.1 Stress

For SLS stress checks, CHBDC poses crack width limitations if the stress exceeds concrete

cracking stress. This study, however, uses a more conservative approach in which cracking is not

permitted during service in order to minimize durability issues. The concrete stress needs to be less

than the tensile strength of concrete in order to avoid cracking:

MSLS

S−

P

A< fcr

where S = sectional modulus P = prestressing force A = gross cross-sectional area

[3-1]

3.4.2.2 Vibration

Vibration limitations are checked according to CHBDC (CSA 2006, Cl.3.4.4) which states that

the deflections under load combination SLS2 must be less than the value described in Figure 3-7.

All analysis cases are assumed to experience frequent pedestrian use which requires the most

stringent vibration criterion.

Figure 3-7. Deflection limits for highway bridge superstructure vibration (CHBDC 2006)

27

3.4.2.3 Deflection

The CHBDC does not have actual deflection restrictions and the limits used in American codes

are optional, because deflections in highway bridges generally do not pose a severe serviceability

problem as in railway bridges. However, excessive deflections affect rider confidence and cause

durability issues such as ponding (MacGregor and Bartlett 2000).

The analysis considers both long-term and short-term deflections due to permanent loads. Long-

term deflection ∆long-term should not exceed 1/750 of the span length (Menn 1990). Any deflection

that exceeds the limit needs to be balanced by camber imposed during construction such that long-

term deflection reduces to zero (Chen and Duan 1999). Long-term deflection accounts for the

minimum upward deflection due to prestress after all losses and the maximum downward deflection

due to creep:

∆long −term = ∆elastic + ∆creep

∆elastic = ∆dead load + ∆prestress after all losses

∆creep = ϕ ∙ ∆elastic

where ϕ = creep coefficient

= 2.0 for CIP

1.5 for precast

[3-2]

On the other hand, short-term deflection accounts for the maximum upward deflection due to

prestress before losses and the minimum downward deflection due to instantaneous dead load:

∆short −term = ∆prestress before losses + ∆instantaneous dead load [3-3]

This check is particularly critical for cases with slender cross-sections in which large prestress

forces can cause the bridge to hog when there is no live load. Excessive camber poses a problem

and cracking might occur at regions that are originally designed to resist compression.

Live load deflections are considered under the vibration requirement. The actual live load

deflections have less impact on rider comfort compared to the acceleration of motion that riders feel

on the bridge.

3.5 Other Preliminary Analysis Assumptions

Other analysis assumptions include:

Tendon eccentricity in duct is neglected. The centre of gravity of tendons is assumed to

be at the centroid of the duct.

The increase in concrete to accommodate intermediate tendon anchors and deviators is

assumed to be negligible compared to the concrete volume of the entire superstructure.

Every span is assumed to have the same number of prestressing tendons. The amount of

prestressing does not vary to suit the demand for each span.

Substructure design is not considered in the analysis.

28

4 ANALYSIS OF CAST-IN-PLACE ON FALSEWORK BRIDGES

This chapter discusses the analysis of cast-in-place on falsework box-girder and solid slab. First,

Section 4.1 examines this construction method. The subsequent sections describe the analysis

models for the two bridge types by considering the cross-sections as well as the prestressing tendon

layouts. Lastly, this chapter summarizes the analysis results including structural behaviours,

vibration limits, deflections, material consumptions, and the factors that restrict further increase of

slenderness ratios.

4.1 Cast-in-Place on Falsework Construction

In cast-in-place on falsework construction, falsework for the entire bridge is assembled first.

The falsework needs to support concrete formwork as well as the full dead load of the bridge during

construction. After concrete is placed, longitudinal internal tendons are installed into ducts and

grout is placed inside the ducts such that tendons become bonded to the concrete. When the

concrete gains sufficient strength, tendons are stressed and as a result, the girder hogs and is

released from the formwork. The formwork and falsework are then removed. This is a labour-

intensive and slow method because it requires not only falsework erection but also on-site

placement of reinforcing steel and concrete. Falsework is also expensive and it can disrupt traffic

below the bridge. Moreover, cast-in-place concrete is subjected to on-site temperature and humidity

changes, so greater effort is needed to ensure good quality compared to precast concrete which is

manufactured in a controlled environment.

Despite these disadvantages, the cast-in-place on falsework method is still used today,

especially in regions where labour is inexpensive, primarily due to its simplicity (Gauvreau 2006).

This flexible construction method can also be applied to bridges with tight curves and complex

geometry (Hewson 2003). Furthermore, it provides opportunities for aesthetic expression since it

can readily accommodate different geometries.

4.2 Cast-in-Place on Falsework Box-Girder

4.2.1 Model

Analysis is performed on 21 cases with interior span lengths of 35m, 50m, 60m, and 75m and

span-to-depth ratios of 10, 15, 20, 25, 30, and 35. This set of span lengths is chosen because cast-in-

place on falsework box-girders are economical for spans up to about 80m according to Menn (1990)

and bridges with longer span lengths need to be haunched in order to reduce dead loads. The end

spans are made 10m shorter than interior spans to balance moments along the entire bridge and to

simplify the treatment of prestressing in the study. If the same span length is used throughout the

29

bridge, the end spans have greater moments than interior spans. A comparison of the moments

obtained from the two span arrangements is shown in Figure 4-1.

Figure 4-1. Moment comparison of bridges with constant span length and reduced end span length

Typical span-to-depth ratios for this type of bridge range from 12 to 35 (Menn 1990). Therefore,

a similar range is used in the analysis. According to Menn (1990), the most economical ratio is 15,

but if both economic and aesthetic impacts are considered, the most optimal range becomes 17 to 22.

4.2.1.1 Cross-Section

A typical cross-section used in the analysis is shown in Figure 4-2. Dimensions for the box

components are based on values suggested by Gauvreau (2006). The deck is 15m wide and supports

four 3.5m wide design lanes and two 0.5m wide barriers. A minimum thickness of 225mm is used

for the top slab. This thickness is assumed to be sufficient to accommodate transverse tendons and

to resist punching shears from wheel loads. The top slab thickness increases to 375mm near the

webs such that the deck slab cantilever has enough strength and stiffness to resist transverse

bending. The intersection of the top slab and web occurs at the quarter point of the deck slab in

order to reduce transverse bending in the web. The web width is 450mm in order to accommodate

reinforcing bars, internal bonded tendons, concrete clear covers and the spacing required to

facilitate concrete placement and vibration. This width remains constant along the entire girder. On

the other hand, the bottom slab thickness varies from a minimum of 200mm at midspan to the depth

of the compressive stress zone created by negative moments at the supports.

Figure 4-2. Typical cross-section for cast-in-place on falsework box-girder

-300

-200

-100

0

100

200

300

0 50 100 150 200 250 300 350 400

Moment (kNm)

Distance (m)

Constant span lengthReduced span length at end spans

30

The reinforcement arrangement illustrated in Figure 4-3 is used and is assumed to be adequate

in resisting transverse bending moments for all analysis cases. The stirrup spacing varies based on

shear requirements; the remaining reinforcing steel is defined as “minimum reinforcing steel” in

this study and the layout is the same for all analysis cases.

Figure 4-3. Typical reinforcing steel layout

4.2.1.2 Prestressing Tendon Layout

The parabolic tendon layout used for this type of bridge is shown in Figure 4-4. The tendon is at

the highest possible location at the supports and the lowest possible location at midspan as allowed

by clear cover requirements of the webs. At the abutments, however, the tendon is located at the

centroidal axis of the cross-section in order to eliminate unbalanced prestress moments. Ideally, the

tendon layout should be parabolic between adjacent supports to provide maximum upward

deviation forces, but this layout results in abrupt corners over the supports. For practical purposes,

the tendon profile needs to be concave downward at the supports and the inflection point occurs at

one-fifteenth of the span length (L/15).

Figure 4-4. Typical tendon profile

In this study, the tendon is assumed to extend from one end of the bridge to another and be

stressed in one operation in order to simulate non-segmental construction. However, this layout is

generally not feasible in actual construction, because the installation is difficult and the prestress

loss due to friction is excessive for long bridges. In a real situation, segmented construction with

staged prestressing is required. This results in a more practical tendon layout in which adjacent

tendons would overlap at intermediate anchors. For the purpose of estimating loss of prestress, the

more practical layout is used and the tendon is assumed to extend over one span only. The

31

overlapping of tendons between adjacent spans is not considered. Based on this layout, the effective

prestress after all losses is estimated to be 60% fpu (1120MPa).

4.2.2 Analysis Results

This section summarizes the analysis results which include the structural response under ULS

and SLS, the material consumptions, and the factors that limit further increase of slenderness ratios.

4.2.2.1 Structural Behaviour and Dimensioning

Table 4-1 describes the ULS strength and SLS stress at the most critical location, the

dimensioning of shear reinforcement, and the factor that determines the amount of prestress

required for each analysis case. The sizing of prestressing tendons is described in Section 4.2.2.4.

Table 4-1. Summary of structural response and dimensioning of cast-in-place on falsework box-girder

L

(m)

L/h

Ultimate limit states Serviceability limit states Governing factor for prestress requirement

__Flexural strength__ __Shear strength__ __Stresses__ MULS

(kNm) Mr

(kNm) MULS/Mr Av

(mm2)

smin (mm)

% of girder @ smin

σSLS (MPa) σSLS/fcr

35

10 37000 38200 97% 1200 - - 0.80 29% ULS flexural

15 35100 35600 99% 2000 - - 0.86 31% ULS flexural

20 33600 34800 97% 2000 243 9.0% 1.07 38% ULS flexural

25 32100 32400 99% 2000 200 22% 1.66 59% ULS flexural

50

10 75100 78400 96% 1200 224 5.4% 1.12 40% ULS flexural

15 67400 68600 98% 1200 147 6.2% 1.91 68% ULS flexural

20 63300 70700 90% 1200 102 17% 1.71 61% SLS stress

25 61200 69900 88% 2000 133 16% 2.10 75% SLS stress

30 60300 73600 82% 2000 106 18% 2.47 88% SLS stress

60

10 115000 118000 98% 1200 219 5.0% 1.16 41% ULS flexural

15 99100 99700 99% 1200 138 8.0% 1.83 65% ULS flexural

20 90200 96500 94% 1200 99 23% 2.01 72% SLS stress

25 85100 92800 92% 1200 76 29% 2.46 88% SLS stress

30 81300 88900 91% 1200 63 33% 2.33 83% SLS stress

35 73500 83300 88% 1200 52 30% 2.45 87% SLS stress

75

10 190000 199000 100% 1200 182 7.2% 1.46 52% ULS flexural

15 161000 164000 98% 1200 112 13% 1.84 66% ULS flexural

20 155000 173000 90% 1200 79 24% 2.10 75% SLS stress

25 142000 147000 97% 1200 59 29% 1.97 70% SLS stress

30 149000 166000 90% 1200 47 37% 2.17 78% SLS stress

35 143000 146000 98% 2000 60 28% 2.09 75% SLS stress

First, under ULS, the table describes the relationship between the flexural strength demand

(MULS) and resistance (Mr). Table 4-1 also summarizes the stirrup requirements for satisfying shear

demand by listing the stirrup area (Av), minimum stirrup spacing (smin) as well as the percentage of

girder that needs stirrups to be placed at a spacing of smin. The remainder of the girder requires

stirrups spaced at every 300mm which is the minimum spacing prescribed by CHBDC (CSA 2006,

32

Cl.8.14.3). All analysis cases, except for the cases with span of 35m and ratios of 10 and 15, need

stirrup spacing of less than 300mm near the supports where shear forces are greatest. Cases with

larger span lengths and span-to-depth ratios require small stirrup spacing of less than 100mm which

should be increased by using a bigger reinforcement bar (i.e. larger than 20M) for construction

purposes. However, the spacing described in Table 4-1 is used in this study, because the required

volume of stirrup varies little with respect to stirrup spacing and it is of greater concern than the

spacing in cost comparisons.

Table 4-1 also compares SLS stresses with the factored cracking stress. SLS stresses (σSLS)

cannot exceed the concrete tensile stress (fcr = 2.8 MPa) to avoid cracking during service.

Lastly, the table shows the factor that governs the amount of prestress needed for each analysis

case. The governing factor is either ULS flexural strength or SLS stress; ULS shear strength does

not govern since stirrups are added to resist any extra shear that is not balanced by the tendons and

concrete. The ULS flexural resistance is proportional to the cross-sectional depth while the SLS

stress depends on the sectional modulus which is related to the second power of the depth. For cases

with low span-to-depth ratios, the sectional moduli are large and the SLS stresses are relatively

small, thus ULS flexural strength governs the prestress requirement. However, as span-to-depth

ratio increases, the sectional modulus decreases at a faster rate than the cross-sectional depth

(Figure 4-5). As a result, SLS stress becomes more critical for slender girders and thus, it governs

the prestress requirement for these cases.

Figure 4-5. Changes in sectional modulus and cross-sectional depth

4.2.2.2 Vibration Limits

Figure 4-6 describes the vibration deflection limits for bridges with frequent pedestrian use and

the truck load deflections under SLS2. The truck deflections are acceptable for all analysis cases

and vibration limits do not govern the prestressing requirement. The truck deflections are at least

58% less than the vibration limits.

Sbottom fibre

h

Stop fibre

0

5

10

15

20

25

30

35

5 10 15 20 25 30 35 40

Sectional modulus S (m3) or depth h (m)

L/h

33

Figure 4-6. Deflection for superstructure vibration limitation

4.2.2.3 Deflections

The following graphs summarize the dead load, long-term and short-term deflections in terms of

span length over deflection (L/∆). As expected, deflection increases with increasing span lengths

and span-to-depth ratios. The maximum camber required is 0.5m for the case with a span of 75m

and span-to-depth ratio of 35 such that the long-term deflection essentially becomes zero.

Figure 4-7. Deflections: a) dead load, b) long-term, c) short-term

L=75m Limit

L=60m Limit

L=50m Limit

L=35m Limit

L=75mL=60m

L=50m

L=35m0

20

40

60

80

5 10 15 20 25 30 35 40

Truck load deflection

(mm)

L/h

L=35m

L=50mL=60mL=75m

0

4000

8000

12000

5 10 15 20 25 30 35 40

Dead loaddeflection

(L/∆)[down]

L/h

L=35m

L=50mL=60mL=75m

0

2000

4000

6000

5 10 15 20 25 30 35 40

Long-termdeflection (creep + elastic)(L/∆)

[down]

L/h

L=35m

L=50m

L=60mL=75m

0

5000

10000

15000

5 10 15 20 25 30 35 40

Short-termdeflection

(dead load+ prestress

before loss)(L/∆)

[down]

L/h

34

4.2.2.4 Material Consumption

Table 4-2 and Figure 4-8 summarize the material consumption for each analysis case as well as

their variations from the baseline case (i.e. L/h=20, which is the conventional ratio defined in

Chapter 2). These results are also illustrated in cross-section drawings in Appendix C.1.

Table 4-2. Summary of material consumptions for cast-in-place on falsework box-girder

L (m)

L/h

Concrete Prestressing tendon Reinforcing steel

Volume (m

3)

% change from baseline case

Number of tendons


Mass (ton)


35

10 2700 +19% 52 -52% 216 +16%

15 2410 +6.5% 76 -30% 193 +3.8%

20 2260 0% 108 0% 186 0%

25 2174 -3.9% 136 +26% 187 +0.5%

50

10 4450 +25% 76 -47% 360 +18%

15 3860 +8.1% 100 -31% 314 +3.2%

20 3570 0% 144 0% 304 0%

25 3420 -4.0% 184 +28% 306 +0.7%

30 3350 -6.0% 240 +67% 315 +0.4%

60

10 5890 +28% 92 -43% 476 +18%

15 5000 +8.9% 120 -25% 410 +1.5%

20 4600 0% 160 0% 404 0%

25 4380 -4.7% 192 +20% 405 +0.2%

30 4270 -7.1% 248 +55% 415 +2.7%

35 4190 -8.8% 300 +88% 416 +3.0%

75

10 8290 +31% 136 -43% 674 +18%

15 6970 +9.9% 184 -23% 585 +2.5%

20 6340 0% 240 0% 571 0%

25 6030 -4.9% 300 +25% 579 +1.4%

30 5850 -7.7% 384 +60% 618 +8.2%

35 5910 -6.8% 528 +120% 621 +8.8%

As shown in the graphs, when the girder becomes more slender, concrete volume decreases

gradually at a decreasing rate. This trend exists because concrete volume depends highly on the

cross-sectional depth which also decreases at a declining rate as span-to-depth ratio increases

(Figure 4-8 a & b). Also, at higher span-to-depth ratios, the concrete volume varies less since the

reduction in web concrete is counterbalanced by the increase in bottom slab thickness. For instance,

for the cases with span length of 50m, concrete volume decreases by 880m3 as span-to-depth ratio

increases from 10 to 20, but concrete volume only decreases by 220m3 as span-to-depth ratio

increases from 20 to 30. In the latter case, the concrete reduction in the webs is 284m3, but the total

concrete reduction is only 220m3 due to increase in bottom slab thickness.

On the contrary, the amount of prestress increases as span-to-depth ratio increases, because

slender bridges have lower flexural resistances and require larger prestress forces (Figure 4-8 c).

35

The number of tendons increases at an increasing rate because for slender cases with large prestress

demand, the tendons need to be placed in more than one layer within the webs, thus lowering the

resistance moment lever arm. The prestress layout becomes more inefficient as slenderness

increases, resulting in an even greater demand for tendons to provide the same level of prestress.

Reinforcing steel mass is attributed to the longitudinal and transverse steel reinforcing bars as

well as the stirrups. The amount of steel reinforcing bars needed is proportional to the cross-

sectional area because they are placed according to Figure 4-3. Consequently, the graph for

reinforcing steel mass follows the trend of concrete volume for cases with low span-to-depth ratios

(Figure 4-8 d). These cases have deep girders that can resist shear using mostly stirrups installed

with the minimum spacing needed to support the tendons (i.e. 20M bars spaced at 300 mm), so the

required amount of reinforcing steel depends on the concrete volume. For the more slender girders,

more stirrups are needed to resist shear and thus the reinforcing steel graph no longer follows the

same trend as the concrete volume graph at higher span-to-depth ratios.

a) Concrete volume

b) Cross-sectional depth

c) Number of prestressing strands

d) Reinforcing steel mass

Figure 4-8. Material consumptions for cast-in-place on falsework box-girder

L=35m

L=50m

L=60m

L=75m

0

2000

4000

6000

8000

10000

5 10 15 20 25 30 35 40

Concretevolume

(m3)

L/h

L=35mL=50m

L=60mL=75m

0.0

2.0

4.0

6.0

8.0

5 10 15 20 25 30 35 40

Depth h (m)

L/h

L=35m

L=50m

L=60m

L=75m

0

100

200

300

400

500

600

5 10 15 20 25 30 35 40

Prestressstrands

L/h

L=35m

L=50m

L=60m

L=75m

0

200

400

600

800

5 10 15 20 25 30 35 40

Reinforcing steel mass

(ton)

L/h

36

4.2.2.5 Limiting Factors of Span-to-Depth Ratios

The upper bound of span-to-depth ratio for cast-in-place on falsework box-girders is restricted

by the number of prestressing tendons that can fit inside the webs and by the minimum height of the

interior box cavity. First, the maximum feasible span-to-depth ratio for cases with span lengths of

75m is limited to 35 due to the tendon arrangement within the webs. This case requires sixteen 37-

strand ducts arranged in four layers in the webs (current tendon arrangement in Figure 4-9). For

cases having ratios beyond 35, more tendons are needed due to the reduced moment lever arm and

section modulus, but they cannot be accommodated efficiently within the 450mm thick webs

according to the current tendon arrangement. This inefficient arrangement of tendons decreases the

eccentricity of tendons, thus further increasing the prestressing demand. Figure 4-9 compares the

current tendon arrangement to a more efficient arrangement in which three ducts are placed in one

layer in each web, thus increasing the eccentricity (i.e. h1 > h2). However, since the web thickness is

kept constant for all cases in this study, the more efficient tendon arrangement is not used. The

current arrangement cannot efficiently accommodate tendons needed for the cases with span lengths

of 75m and slenderness ratios beyond 35.

Figure 4-9. Tendon arrangement that limits further increase in span-to-depth ratio

If the more efficient tendon arrangement from Figure 4-9 is used, the prestressing demand is

expected to decrease. For example, for the case with span length of 75m and ratio of 35, using the

more efficient arrangement increases the eccentricity by 89mm and decreases the prestressing

demand by 3.0%. This decrease in tendon is relatively minor, because the advantage from

additional tendon eccentricity is offset by the increase in dead load (i.e. concrete volume increases

by 4.3% due to the thicker webs). With thicker webs, the maximum feasible span-to-depth ratio for

cases with span lengths of 75m becomes 40 at which point the limiting factor is the minimum

height requirement of the interior box cavity. Therefore, the results of this analysis depend on the

assumed web thickness. Using thinner webs requires more prestressing tendons and reduces the

maximum feasible span-to-depth ratio while larger web thickness reduces prestressing demand and

expands the feasible range of ratios.

More efficient tendon arrangement Current tendon arrangement

h2

h1

37

Another factor that restricts span-to-depth ratios is the minimum height requirement for the

interior box cavity. As shown in Figure 4-10, the interior box cavity needs to be at least 1.0m such

that workers have enough space to strip forms, stress tendons and perform maintenance and repairs.

Figure 4-10. Interior box cavity limitation

Although this height limit is not required by CHBDC, it is often used as good construction

practice. This height requirement restricts the ratio for cases with span lengths of 35m, 50m, and

60m to 25, 30, and 35 respectively. As span-to-depth ratio increases, more tendons are used and the

bottom slab becomes thicker in order to accommodate the compressive stress zone in negative

moment regions. As the bottom slab thickness increases, the height of interior box cavity shortens.

If this minimum height restriction does not exist, the box would essentially turn into a solid slab for

the more slender cases (Figure 4-11).

Figure 4-11. Height of access diminishes as span-to-depth ratio increases

heightof

accessh

38

4.3 Cast-in-Place on Falsework Solid Slab

As shown in the previous analysis, box-girders are not feasible for slender cases (i.e. span-to-

depth ratio greater than 30) due to practical construction considerations. Any further reduction in

cross-sectional depth essentially turns the box into a solid slab. This finding leads to the analysis of

cast-in-place on falsework solid slabs described in this section.

From the aspect of construction cost optimization, investigating span-to-depth ratios for solid

slabs is valuable, because concrete savings as a result of reduced depth in solid slabs would be more

than those in box-girders. As indicated by the shaded areas in Figure 4-12, reducing depth in solid

slab eliminates a strip of concrete as wide as the soffit while reducing depth in box-girder only

removes the web concrete.

Cast-in-place on falsework solid slabs are economical for short-span bridges due to simple

formwork, prestress layout and concreting operations (Hewson 2003). The straightforward

construction does not require a high level of technology or an extensive amount of labour. However,

solid slabs are inefficient in terms of structural behaviour, because they result in relatively large

dead loads and need more prestressing to get sufficient flexural stiffness and resistance compared to

box-girders (Menn 1990).

Due to its excessive dead load, solid slabs are generally used for shorter spans of less than 20m

(Gauvreau 2006). To reduce the dead load for long span cases, voided slabs (Figure 4-13) are used

instead of solid slabs. In voided slabs, stay-in-place forms are used to create the hollow cores. These

forms must be anchored against uplift during concreting and they need vents and drainage openings,

thus complicating the construction process. The voids also pose durability issues since inspection is

not possible inside the voids. Due to the construction complications and durability concerns, voided

slabs are not considered in the analysis despite the savings in concrete and reinforcing steel.

Figure 4-13. Voided slab

Figure 4-12. Concrete reduction due to increase in span-to-depth ratio for solid slab and box-girder

39

4.3.1 Model

The analysis considers 18 cases with span lengths of 20m, 25m, 30m, and 35m and span-to-

depth ratios of 30, 35, 40, 45, and 50. According to Gauvreau (2006), the maximum cost-effective

slab depth is 0.8m and the maximum practical span-to-depth ratio is 25, resulting in a maximum

span of 20m. This analysis uses a higher range of spans and ratios in order to demonstrate the

impacts of slenderness. Bridges with spans shorter than 20m cannot achieve the proposed span-to-

depth ratios because the girders would be too slender to accommodate sufficient reinforcement (e.g.

a 15m long span with a ratio of 35 only has a depth of 0.43m). Also, this set of ratios is chosen

since the commonly used value is 30 based on the review of existing bridges in Chapter 2.

Furthermore, span-to-depth ratios below 30 are not investigated, because a deep cross-section with

a depth greater than approximately 0.8m has large dead loads and is not economical (Menn 1990).

4.3.1.1 Cross-Section

A typical cross-section and reinforcement layout are shown in Figures 4-14 and 4-15. Compared

to the box cross-section considered previously, the solid slab model has a wider deck (i.e. 22m) that

supports six design lanes and shorter deck cantilevers (i.e. 3.75m) in order to emphasize slab

behaviour. Also, the deck cantilevers are tapered to reduce dead load which is much higher in solid

slabs than in box-girders. Longitudinal internal bonded tendons are grouped in bands over support

lines based on recent practices in North America (Park and Gamble 2000). To simplify analysis, a

solid wall pier is assumed and thus, only one band of tendons that spread over the entire “spine” of

the cross-section is used. To carry loads into the longitudinal tendon band, transverse tendons are

uniformly distributed along the length of the bridge.

Figure 4-14. Typical cross-section for cast-in-place on falsework solid slab


4.3.1.2 Prestressing Tendon Layout

The model has the same tendon layout as the one from the box-girder model (Figure 4-4).

40

4.3.2 Strip Method versus Beam Model

Designing reinforcement based on elastic sectional forces only is complicated in slabs, because

bending moments exist in two orthogonal directions. To simplify the design, two methods are

considered: strip method and beam model. Results from these two methods are compared in this

section for three analysis cases: 1) L=25m, L/h=30; 2) L=30m, L/h=40; 3) L=35m, L/h=50.

The strip method, proposed by Hillerborg (1996), generates lower bound solutions of the theory

of plasticity. In contrast to yield line theory, the strip method solution provides adequate flexural

safety at ULS. This method states that if the moments can be distributed such that equilibrium

equations are satisfied and the reinforcements are designed for these moments, then the slab is safe

at ULS. The solution only needs to fulfill equilibrium equations and not necessarily the

compatibility criteria. More than one solution is possible since the slab is statically indeterminate.

For design purposes, the solution that yields reinforcement economy and favorable behaviour under

service conditions is used.

In the simple strip method, load is assumed to be carried by strips that run in the longitudinal

and transverse reinforcement directions. These strips are treated as beams and the moment in each

strip can be solved using simple statics. To further simplify analysis, torsional moments are

assumed to be zero in these strips. This moment distribution is preferred because torsional moments

require more reinforcement. One way to apply the simple strip method in prestressed slabs is to

divide the slab into strips that contain one tendon each. Since all the tendons are equal, strip width is

varied in order to balance the load.

Menn (1990) proposed a straightforward way to redistribute moments in slabs. In order to

minimize cracking, the prestressing deviation forces (qp) are chosen to be 60-80% of the total dead

load. To simplify analysis and construction, parallel prestressing tendons are arranged into narrow

bands. These bands of tendons are idealized as beams in the longitudinal direction and slabs can

redistribute moments onto the bands to reduce peak stresses. To account for this moment

redistribution onto these reinforcement bands, a self-equilibrating moment (msp) is used:

msp = mp − m0p

where mp = moment where qp is applied to the original slab

-m0p = moment when-qp is applied to the idealized slab in which

the prestressing bands act as individual beams

[4-1]

This self-equilibrating moment varies across the width of the cross-section since additional

positive moments are concentrated at the reinforcement bands. Reinforcements are designed for the

final moments after redistribution.

Design moments in the analysis are computed based on Menn’s prestressing concept. First, the

structure is divided into two 3.75m wide edge strips and one 14.5m wide prestress band as shown in

41

Figure 4-16 b. Longitudinal tendons are distributed evenly in the prestress band. The two edge

strips are narrow relative to the prestress strip in order to ensure the deck cantilever load would

travel into the prestress band due to the short load path. To compute –m0p, - qp b

b0 is applied to the

prestress band where qp is 80% of the total dead load, b is the total slab width and b0 is the width of

prestress band. Likewise, mp is obtained by applying qp to the entire slab. The redundant moment

(msp) is the difference between mp and m0p. The redundant moment transfers forces from the edge

strips to the prestress band. Since this moment is self-equilibrating, the total moment remains the

same after redistribution as shown in Figure 4-16 g. Reinforcement is then designed such that the

following is satisfied the factored moment (Figure 4-16 h):

αDL mDL + αLL mLL + αpmsp < Mr

where α=load factor Mr=moment resistance

[4-2]

The transverse distribution of longitudinal bending moment across the slab and the final design

moment are illustrated in Figure 4-16.

a) Elevation:

b) Section A-A:

c) Dead load and live load moments mDL + mLL (assume mDL = mLL):

d) -m0P due to application of deviation forces qp on prestress band:

e) mP due to application of deviation forces qp on the entire slab width:

f) Self-equilibrating moment mSP = -m0P + mP:

g) Unfactored design moment = mDL + mLL + msp:

h) Design moment = αDLmDL + αLLmLL + αpmsp:

Figure 4-16. Transverse distribution of longitudinal bending moment in slabs (values are divided by mDL)

Edge strip Prestress band b0= 14.5m Edge strip

b = 22m

2.00

1.21

0.80

0.80

0.41

1.20

2.41 mDL

+mLL

= 2.00

3.292.14

A

A

Support Support


b = 22m

2.00

1.21

0.80

0.80

0.41

1.20

2.41 mDL

+mLL

= 2.00

3.292.14

A

A

Support Support


b = 22m

2.00

1.21

0.80

0.80

0.41

1.20

2.41 mDL

+mLL

= 2.00

3.292.14

A

A

Support Support


b = 22m

2.00

1.21

0.80

0.80

0.41

1.20

2.41 mDL

+mLL

= 2.00

3.292.14

A

A

Support Support


b = 22m

2.00

1.21

0.80

0.80

0.41

1.20

2.41 mDL

+mLL

= 2.00

3.292.14

A

A

Support Support


b = 22m

2.00

1.21

0.80

0.80

0.41

1.20

2.41 mDL

+mLL

= 2.00

3.292.14

A

A

Support Support


b = 22m

2.00

1.21

0.80

0.80

0.41

1.20

2.41 mDL

+mLL

= 2.00

3.292.14

A

A

Support Support


b = 22m

2.00

1.21

0.80

0.80

0.41

1.20

2.41 mDL

+mLL

= 2.00

3.292.14

A

A

Support Support


b = 22m

2.00

1.21

0.80

0.80

0.41

1.20

2.41 mDL

+mLL

= 2.00

3.292.14

A

A

Support Support

42

The three analysis cases are also investigated under the assumption that the slab behaves like a

beam, so there is no moment redistribution. The longitudinal tendon requirements for ULS flexure

determined from the two methods are summarized in Table 4-3.

Table 4-3. Results from beam model and strip method

L=25m, L/h=30 L=30m, L/h=40 L=35m, L/h=50

Beam Strip % diff. Beam Strip % diff. Beam Strip % diff.

Tendon requirement 230 176 23% 360 351 2.5% 500 486 2.8% Total construction cost per deck area $ 2170 $ 2110 2.8% $ 2480 $ 2540 2.4% $ 2580 $ 2570 0.4%

The table shows that the prestress requirements and total construction costs obtained from the

two methods are similar. Strip method does not change the results significantly because the analysis

model is predominantly a simple, one-way system, so the moment distribution is similar to that of a

beam. Therefore, the simpler beam assumption is used for the analysis.

4.3.3 Analysis Results

4.3.3.1 Structural Behaviour and Dimensioning

Table 4-4 describes the ULS strengths and SLS stress at the most critical location and the

dimensioning of shear stirrups for each analysis case. The sizing of prestressing tendon is described

in Section 4.3.3.5. For all cases, the ULS flexural resistance (Mr) is at least 27% greater than the

demand (MULS), so flexural strength does not govern the prestress requirement. However, to satisfy

the maximum reinforcement criterion, larger concrete compressive strengths (f’c) are needed. This

criterion is discussed in greater details in Section 4.3.3.2. On the other hand, shear strength

requirements can be fulfilled with the listed stirrup areas (Av) distributed at a minimum spacing of

300mm as described in CHBDC. Only minimum stirrups are needed because the cross-sectional

widths and concrete strengths are large, thus resulting in high concrete shear resistances. Lastly,

SLS stresses govern the prestressing requirement for all cases. Critical stress occurs at the top fibre

of the support which is subjected to large negative moments.

43

Table 4-4. Summary of structural response and dimensioning of cast-in-place on falsework solid slab

L (m)

L/h

Ultimate limit states Serviceability limit states

__Flexural strength__ _Shear strength_ __Stresses__ MULS (kNm) Mr (kNm) MULS/Mr f’c (MPa) Av (mm

2) σSLS (MPa) σSLS/fcr

20

30 17500 20000 86% 50 600 2.50 89%

35 15900 19100 83% 50 600 2.63 93%

40 14800 18900 78% 75 600 3.04 88%

25

30 28600 34000 84% 50 600 2.54 90%

35 25500 31600 81% 50 1000 2.42 85%

40 23400 30000 79% 60 1000 2.64 85%

30

30 43900 49600 89% 50 600 2.67 94% 35 39300 45800 86% 50 1000 2.75 97%

40 36500 44300 82% 60 1000 3.03 98%

45 32800 42900 76% 75 1000 3.03 87%

35

30 68900 81100 85% 50 600 2.79 98% 35 61200 74300 82% 50 1000 2.54 90%

40 55900 68100 82% 60 1000 2.92 94%

45 51800 64100 81% 80 1000 3.48 97%

4.3.3.2 Maximum Reinforcement Criterion

According to the maximum reinforcement criterion in CHBDC (CSA 2006, Cl.8.8.4.5), the

flexural resistance at ULS should be developed with a c/d of less than 0.5 at regions where the

moment capacity is close to the demand (c is the height of compressive stress region while d is the

distance between the extreme compressive fibre and the tendon as shown in Figure 4-17). This

ensures that the tendons would already be yielding and significant plastic deformations have

occurred when concrete crushes such that the structure would fail in a ductile manner. This

limitation is not satisfied for the more slender analysis cases. Slender bridges have short moment

lever arms and thus large prestress requirements. A large tensile force in the tendon needs to be

balanced by an equally large concrete compressive force, meaning that c is large. Slender bridges

have small d due to physical limitation and large c due to moment requirements, resulting in large

c/d ratios. For cases that do not satisfy this criterion, concrete strengths are increased such that

prestress requirement and c decrease. Concrete strengths that are needed to satisfy the maximum

reinforcement limit are summarized in Table 4-5. These higher strength concretes are more

expensive and the additional expenses will be accounted for in the cost comparative studies in

Chapter 6. For the cases that are crossed out in the table, concrete with strengths greater than 80

MPa are needed to fulfil the criterion. Such high strengths are not widely used in today’s bridge

industry and hence, the maximum span-to-depth ratio is limited by the maximum reinforcement

criterion and the maximum practical concrete strength.

44

Figure 4-17. Maximum reinforcement criterion: a)

concrete stains, b) equivalent concrete stresses, c) concrete

forces

Table 4-5. Concrete strengths required to satisfy

maximum reinforcement criterion

f’c required to satisfy c/d<0.5 limit (MPa)

L/h 20m 25m 30m 35m

30 50 50 50 50

35 50 50 50 50

40 75 60 60 60

45 150 90 75 80

50 ― ― 135 115

4.3.3.3 Vibration Limits

Figure 4-18 describes the vibration deflection limits for bridges with frequent pedestrian use

and the truck load deflections under SLS2. The truck deflections are acceptable for all analysis

cases and vibration limits do not govern the prestressing requirement. The truck deflections are at

least 52% less than the vibration limits. The deflection increases as the girder becomes more slender

until the span-to-depth ratio reaches 35 for span lengths of 20m and 25m, and 40 for span lengths of

30m and 35m. These cases use larger concrete strengths as described previously (Section 4.3.3.2),

resulting in greater stiffness as well as lower deflections and flexural frequencies (indicated by

triangular markers on Figure 4-18).


L=35m limit

L=30m limit

L=25m limit

L=20m limit L=35mL=30mL=25m

L=20m

0

10

20

30

40

50

25 30 35 40 45 50

Truck load deflection

(mm)

L/h

c

εcu

= -3.5x10-3

d

∆εp

fp

0.85f’c

T

C

z

45

4.3.3.4 Deflections

The following graphs summarize the dead load, long-term and short-term deflections in terms of

span length over deflection (L/∆). The maximum camber required is 0.075m such that the long-term

deflection essentially becomes zero for the case with a span of 35m and slenderness ratio of 40. As

explained previously, cases with increased concrete strengths have lower deflections; these cases

are indicated by triangular markers.


L=35mL=30m

L=25m

L=20m

0

4000

8000

12000

25 30 35 40 45 50

Dead load deflection

(L/∆)[down]

L/h

L=35mL=30m

L=25mL=20m

0

4000

8000

12000

25 30 35 40 45 50

Long-term deflection (creep + elastic) (L/∆)

[down]

L/h

L=20mL=25m

L=30mL=35m

0

10000

20000

30000

25 30 35 40 45 50

Short-term deflection

(dead load + prestress

before loss) (L/∆)[up]

L/h

46

4.3.3.5 Material Consumption

Table 4-6 and Figure 4-20 summarize the material consumption for each analysis case and its

variation from the baseline case (i.e. L/h=30, which is the conventional ratio defined in Chapter 2).

The cases with higher concrete strengths are italicized in the table and are indicated by triangular

markers in the graphs. These results are also illustrated in cross-section drawings in Appendix C.2.

Table 4-6. Summary of material consumption for cast-in-place on falsework solid slab

L (m)

L/h


Volume (m

3)


Number of tendons


Mass (ton)


20

30 2150 0% 210 0% 85.7 0%

35 1930 -10% 264 +26% 86.9 +1.4%

40 1770 -18% 322 +53% 88.0 +2.7%

25

30 3210 0% 260 0% 111 0%

35 2860 -11% 312 +20% 113 +1.7%

40 2600 -19% 364 +40% 114 +2.5%

30

30 4270 0% 300 0% 124 0%

35 3790 -11% 350 +17% 130 +4.8%

40 3430 -20% 414 +38% 131 +5.4%

45 3160 -26% 480 +60% 132 +5.8%

35

30 5710 0% 414 0% 148 0%

35 5050 -12% 480 +16% 156 +5.0%

40 4550 -20% 528 +28% 156 +5.2%

45 4170 -27% 576 +39% 156 +5.3%

Contrary to the non-linear decline in box-girders, concrete volume decreases at a relatively

steady rate as slenderness increases for solid slabs (Figure 4-20 a). In box-girders, concrete volume

diminishes at a decreasing rate since web concrete reduction is offset by bottom slab thickening. In

contrast, increasing span-to-depth ratio in solid slabs simply removes an entire strip of slab, so the

concrete volume reduction is directly proportional to the decrease in girder depth (Figure 4-20 b).

The amount of prestressing is expected to increase at an increasing rate as slenderness rises

based on observations described in Section 4.2.2.4 for box-girder. However, Figure 4-20 c shows

that the prestress increases linearly with slenderness since larger concrete strengths are used and

fewer tendons are needed for the slender cases. Also, solid slabs can accommodate a large number

of tendons at the same elevation, so the tendons do not need to be placed in layers which reduce the

prestressing efficiency.

As shown in Figure 4-20 d, the amount of reinforcing steel remains relatively constant as span-

to-depth ratio varies. For the cases with spans of 20m, the stirrup requirement is the same for all

three cases. For the cases with the other span lengths, as span-to-depth increases from 30 to 35, the

required stirrup area increases from 600mm2 to 1000mm

2 but spacing remains at 300mm. This

47

increase in stirrup area has little effect because stirrups account for less than 10% of the total

amount of reinforcement steel. As a result, the amount of reinforcement steel does not deviate more

than 5% from the baseline case.

a) Concrete volume

b) Cross-sectional depth

c) Number of prestressing strands

d) Reinforcing steel mass

Figure 4-20. Material consumptions for cast-in-place on falsework solid slab

4.3.3.6 Limiting Factors of Span-to-Depth Ratios

Maximum span-to-depth ratio in solid slabs is limited by the maximum reinforcement criterion

which states that the height of the compressive stress region should be less than half the distance

from the extreme compressive fibre to the centroid of tensile force (i.e. c/d < 0.5). This criterion is

not satisfied for cases with ratios as low as 35 if a concrete strength of 50 MPa is used. In order to

fulfil the criterion, cases with higher ratios use larger concrete strengths which reduce prestress

demands (i.e. results in a smaller compressive stress region) and improve ductility. The maximum

concrete strength used in the analysis is 80 MPa which limits the span-to-depth ratio to 40 for cases

with spans of 20m and 25m, and to 45 for cases with spans of 30m and 35m.

L=20m

L=25m

L=30m

L=35m

0

2000

4000

6000

25 30 35 40 45 50

Concretevolume (m^3)

L/h

L=35m

L=30mL=25m

L=20m

0

0.4

0.8

1.2

25 30 35 40 45 50

Depth (m)

L/h

L=20mL=25m

L=30m

L=35m

0

200

400

600

800

25 30 35 40 45 50

Prestressstrands

L/h

L=20m

L=25m

L=30m

L=35m

0

50

100

150

200

25 30 35 40 45 50

Reinf. steelmass(ton)

L/h

48

5 ANALYSIS OF PRECAST SEGMENTAL SPAN-BY-SPAN BOX-

GIRDER

This chapter examines the optimization of span-to-depth ratio for precast segmental span-by-

span constructed box-girders. Section 5.1 provides a general description of the construction method;

Section 5.2 discusses the analysis model; Section 5.3 examines the construction moments and

moments due to thermal gradients; Section 5.4 explores the loss of prestress; Section 5.5 discusses

the behaviour of unbonded tendons at ultimate limit states; and Section 5.6 summarizes the analysis

results.

5.1 Precast Segmental Span-by-Span Construction

Precast segmental span-by-span construction method is ideal for long bridges with many short-

to-medium spans, because it has a high speed of construction when repetitive spans are used. For

instance, a typical 40m span can be erected every 2 to 3 days using this method (Hewson 2003).

Also, this method can be used for bridges that require access beneath the superstructure since the

erection equipment is mainly above ground and the disruption to traffic below the bridge is

minimized (Sauvageot 1999).

The construction sequence is illustrated in Figure 5-1. In this method, an erection girder, either

an overhead truss or underslung girder, is first supported on a previously completed deck or

adjacent piers. Then precast segments are transported from the completed span and a crane located

at the edge of the completed deck stacks the segments loosely on the erection girder. If an overhead

truss is used, the segments are hanged with cables (Figure 5-2 a). If an underslung girder is used,

segments are supported under the deck cantilevers or soffit (Figure 5-2 b). After all the segments of

an entire span are placed, they are connected together with temporary prestress. A cast-in-place

closure joint next to the pier segment stitches the new span with the previously completed span

(Hewson 2003). Then permanent longitudinal external tendons, which overlap the previous span,

are installed and the new span becomes continuous with the completed structure. The erection

girder then launches forward to erect the next span.

Since precast segments are used and tendons are only stressed once for the entire span, this

method increases the ease and speed of construction. Efficiency in construction is further enhanced

with the use of external tendons which increases the speed of precasting (Sauvageot 1999).

49

Figure 5-1. Precast segmental span-by-span construction method

Figure 5-2. Span-by-span erection girder: a) overhead truss, b) underslung girder (NRS 2008, OSHA 2006)

5.2 Model

Analysis is performed on 11 cases with span lengths of 30m, 40m and 50m and span-to-depth

ratios of 15, 20, 25, and 30. This set of span lengths is chosen based on lengths of erection girders

that are typically used in the industry today. The maximum length is 50m because longer erection

equipment would become too heavy and this method would no longer be economical (Sauvageot

1999).

The range of slenderness ratios chosen for analysis is based on the conventional optimal ratio of

17 and the minimum cross-sectional depth of 1.8m (Gauvreau 2006). This depth is larger than the

minimum depth used in cast-in-place on falsework box-girder analysis (i.e. 1.4m) since the precast

bridge is longitudinally post-tensioned with external unbonded tendons which shorten the moment

lever arm compared to internal tendons.

Erection girderPrecast segments are placed onto erection girder

Prestressing tendons are installed

Erection girder launches forward

Direction of ConstructionErection girder

Precast segments are placed onto erection girder



Direction of Construction Erection girderPrecast segments are placed onto erection girder



Direction of Construction

50

5.2.1 Cross-Section

A typical cross-section and reinforcement layout are shown in Figures 5-3 and 5-4. They are

modified from AASHTO-PCI-ASBI standard sections (1997). Except for box height and bottom

slab thickness, all other dimensions remain the same for every analysis case. The box webs are

more slender compared to those for cast-in-place on falsework because they do not need to

accommodate internal tendons. As a result, web thickness depends only on shear and transverse

bending requirements. The bottom slab thickness depends on the thickness of the compressive stress

zone at negative moment regions. The deck is 15m wide and can support four lanes of traffic which

is the same as the cast-in-place on falsework case.

Figure 5-3. Typical cross-section for precast segmental span-by-span box-girder


5.2.2 Elevation and Prestressing Tendon Layout

In precast segmental span-by-span construction, the girder is composed of interior segments,

pier segments, and closure joints. The maximum length of interior segment is chosen to be 3m

based on limitations of transporting segments from casting yard to construction site. The pier

segments are 1m long and they support the anchorages of all longitudinal tendons. The 150mm long

51

cast-in-place closure joint fills the gap between the pier segments and interior segments and it

allows for adjustments during placement of segments. Elevation and typical tendon layout of an end

span is illustrated below.

Figure 5-5. Typical tendon profile

As shown in the diagram, longitudinal tendons are continuous between adjacent piers and they

connect the precast segments of each span. Since the tendons are externally unbonded, a straight

tendon profile is used instead of a parabolic profile. Each tendon begins at the centroidal axis of the

girder for end spans or directly below the top slab for the interior spans. The tendon then continues

within the box cavity towards the bottom slab. It is deviated at 1/3 of the span and assumes a

horizontal profile. When the tendon reaches 2/3 of the span, it deviates again and continues towards

the top slab of the pier segment. The tendon is then anchored at the exterior side of the pier segment.

This anchorage location allows tendons from adjacent spans to overlap at the pier segments, thus

providing continuity throughout the superstructure.

5.3 Longitudinal Bending Moments

This section discusses two types of longitudinal bending moments that are particularly critical

in precast segmental bridges: construction moments and thermal gradient moments.

5.3.1 Construction Moments

Because a segmental method is used, the structural system changes during construction and the

final moments are affected by the stress history. Therefore, the moments at each stage of

construction is computed to obtain the final moments at completion. The following illustration

describes the dead load and prestressing moments (MDL and MP,tot) for a case with span of 40m and

ratio of 20 at various construction stages. The first eight graphs show the moments caused by the

additional new span at each construction stage. The last graph shows the final moments at the

instant of completion which are obtained by summing the construction moments at all construction

stages.

52

Figure 5-6. Construction moments for segmental span-by-span method

Stage 1: Stage 2:

Stage 3: Stage 4:

Stage 5:

MDL

MP,tot-40000

-20000

0

20000

40000

0 40

Moments (kNm)

Distance (m)

MDL

MP,tot

-40000

-20000

0

20000

40000

0 40 80

Moments (kNm)

Distance (m)

MDL

MP,tot

-40000

-20000

0

20000

40000

0 40 80 120

Moments (kNm)

Distance (m)

MDL

MP,tot-40000

-20000

0

20000

40000

0 40 80 120 160

Moments (kNm)

Distance (m)

MDL

MP,tot

-40000

-20000

0

20000

40000

0 40 80 120 160 200

Moments (kNm)

Distance (m)

53

Figure 5-6. Construction moments for segmental span-by-span method (continued)

Stage 6:

Stage 7:

Stage 8:

Final moments at the instant of completion: The final moments are the sum of all the construction moments.

MDL

MP,tot-40000

-20000

0

20000

40000

0 40 80 120 160 200 240

Moments (kNm)

Distance (m)

MDL

MP,tot

-40000

-20000

0

20000

40000

0 40 80 120 160 200 240 280

Moments (kNm)

Distance (m)

MDL

MP,tot

-40000

-20000

0

20000

40000

0 40 80 120 160 200 240 280 320

Moments (kNm)

Distance (m)

MDL

MP,tot

-40000

-20000

0

20000

40000

0 40 80 120 160 200 240 280 320

Moments (kNm)

Distance (m)

54

In the span-by-span method, the dead load moments at the instant of completion are biased

towards positive moment due to stress history compared to the same bridge built on falsework.

However, the instantaneous span-by-span moments (MSBS) would redistribute over time and shift

towards falsework moments (MFAL) because of creep. This redistribution of moments is

approximated by the following formula (Menn 1990):

M∞ = MSBS + α(MFAL + MSBS ) [5-1]

The factor α is chosen to be 0.7 for precast concrete. If cast-in-place concrete is used, α would

be 0.8 since it is more susceptible to effects of creep than precast concrete. The following graph

shows the redistribution of dead load moments for a case with L=40m and L/h=20.

Figure 5-7. Redistribution of dead load moments due to creep

When considering dead load moment only, significant redistribution of moments causes the

moment demand at midspan to decrease and demand at supports to increase. However, prestress

moments should also be included in the redistribution as shown in the following graph.

Figure 5-8. Redistribution of dead load and prestress moments due to creep

As illustrated in the graph, redistribution of the sum of prestress and dead load moments is

negligible. Therefore, moment redistribution is not considered in the analysis and the instantaneous

moments are used as moment demands for ULS and SLS checks.

5.3.2 Moments due to Thermal Gradient

For heavy bridges exposed to solar radiation, thermal gradients exist across the cross-section

depths, meaning that the top deck is warm while the bottom flange is cold. As a result, the girder

deflects upward under no external loads.

-40000

-20000

0

20000

40000

0 40 80 120 160 200 240 280 320Dead load moment

(kN)

Distance (m)

Falsework Span-by-span M at infinity Moment at infinity

-40000

-20000

0

20000

40000

0 40 80 120 160 200 240 280 320Dead load +prestressingmoments

(kN)

Distance (m)

Falsework Span-by-span M at infinity Moment at infinity

55

In a simply-supported girder, thermal gradient only causes deflection and not moments. On the

other hand, it imposes a moment on continuous girders since there are internal restraints. These

thermal gradient moments are directly proportional to the flexural stiffness. Therefore, thermal

gradient is a particularly important issue for span-by-span constructed bridges in which cross-

sections are generally stiffer. Thermal gradient causes positive moments, which can crack the

bottom flange, so additional bottom tendons are required in order to prevent such cracking. Also,

thermal gradient moments are considered only in serviceability design checks, because the cracked

flexural stiffness is small in ultimate limit states and thus thermal gradient moments are negligible.

In the analysis, thermal gradient effects are considered for serviceability limit states and are

obtained using the method described as follows. First, the continuous girder is made statically-

determinate by removing internal restraints. A thermal differential θ(y) of 10°C is then applied

between the 225mm thick top deck and the rest of the box (CSA 2006, Cl.3.9.4.4). The free strain

due to this temperature change is obtained by the following formula:

εf = αcθ(y) where αc = thermal coefficient of concrete

= 10 × 10−6/℃ [CSA 2006, Cl.8.4.1.3]

[5-2]

This free strain profile follows the shape of the temperature change and is non-linear if

unrestrained expansion is allowed at all elevations. However, the final strain profile ε(y) needs to be

linear since plane section remains plane. The difference between the free strain and final strain

profiles gives the restraint strain which is needed to restore compatibility. The thermal gradient and

strain profiles are illustrated in the following diagram.

Figure 5-9. Thermal gradient effects

These self-equilibrating, restraint stresses due to non-linear thermal profile on a statically-

determinate structure causes the primary temperature stresses fr(y):

fr y = Ec ε y − εf = Ec ε y − αcθ y

[5-3]

56

The axial restraint force Pr and restraint moment Mr can then be computed from fr (y):

Pr = fr y b y dyh

0

= Ec ε y − αcθ y h

0

b y dy [5-4]

Mr = fr y b y y − n dyh

0


0

b y y − n dy [5-5]

However, both Pr and Mr are equal to zero because there are no external forces and internal

redundancies have been removed to make the structure statically-determinate. By setting Equations

5-4 and 5-5 to zero, the final curvature of bending (ψ) and final strain (ε0) at elevation y=0 are

obtained as follows. A complete derivation is shown in Appendix B.3.

ψ =αc

I θ y b y y − n dy

h

0

[5-6]

ε0 =αc

A θ y b y dy

h

0

− nψ [5-7]

According to the final strain profile, the final strain at elevation y can be obtained by:

ε y = ε0 + ψy [5-8]

As a result, the primary thermal stresses become:

fr y = Ec ε0 + ψy − αcθ(y) [5-9]

In a statically-determinate structure, these primary stresses only cause the girder to hog, but

there is no net moment. However, in a continuous structure, the curvature Ψ created by primary

stresses is incompatible and are restrained by piers. In order for the structure to maintain

compatibility while a thermal deformation is imposed, positive restraint moments are needed at

each end of the span:

M = -EcIψ [5-10]

The final moment Mfinal is obtained using moment distribution (Figure 5-10). This positive

restraint moment causes tensile stresses at the bottom fibre near the piers where the post-tensioning

are arranged to resist negative moments. If the positive restraint moments are large enough to crack

the bottom flange, additional bottom tendons would be required near the piers such that the resultant

moments become negative.

Figure 5-10. Moments due to thermal gradient

0

2000

4000

6000

0 40 80 120 160 200 240 280 320

Moments due to thermal gradient

(kNm)

Distance (m)

57

Furthermore, secondary thermal stresses fs are induced by Mfinal:

fs y =Mfinal (y − n)

I

[5-11]

The total thermal stresses that need to be considered in design checks are computed with the

following equation:

ftemp y = fr y + fs y

= Ec ε0 + ψ ∙ y − αcθ(y) +Mfinal (y − n)

I

[5-12]

5.4 Loss of Prestress

Since external, unbonded tendons are used, the loss of prestress due to friction is lower and the

effective prestress after all losses is expected to be greater than 60%fpu (1120 MPa) which is the

estimated value used for the cast-in-place on falsework analyses. Therefore, an explicit calculation

of prestress losses is performed. With a more detailed calculation, the increase in prestress is less

than 6% compared to the assumed value of prestress after all losses (1120 MPa = 60% fpu). This

increase in stress is not significant enough to reduce the prestressing requirement.

5.4.1 Friction Losses

Friction losses are proportional to the deviation angles of tendons. Thicker sections have larger

deviation angles and thus higher friction losses. The decrease in prestressing force due to friction

losses ∆P(x) are computed with the following formulae (Menn 1990).

∆P x = P0 1 − e−μα x if μα x ≥ 0.2

P0μα x if μα x < 0.2

where x=distance from stressing location

P0=jacking force σp0=jacking stress=80%fpu=1490MPa

μ=coefficient of friction=0.25 for external ducts α x =sum of angle changes between stressing location and point x

=αx+x∆α αx=intentional angle changes refer to Figure 5-11

=α1+α2+…+αn ∆α=unintentional angle change due to construction tolerances and

displacement of tendon during concreting =0° for external tendons

[5-13]

Figure 5-11. Intentional angle changes

58

The friction losses are summarized in the Table 5-1. The maximum loss is only 5% of fpu.

Table 5-1. Prestress losses due to friction

Friction losses (% of fPu)

L/h 30m 40m 50m

15 4.33% 4.91% 5.09%

20 2.70% 3.30% 3.56%

25 − 2.37% 2.62%

Table 5-2. Prestress losses due to anchorage set

Anchorage set losses (% of fPu)

L/h 30m 40m 50m

15 6.11% 5.63% 5.12%

20 4.68% 4.62% 4.29%

25 − 3.76% 3.68%

The loss of prestress due to anchorage set is a function of friction losses. The anchorage set

is assumed to be 6mm and the prestress loss due to anchorage set is computed using Equation 5-14.

(Collins and Mitchell 1997). The prestress losses due to anchorage set are summarized in Table 5-2.

∆P x = 2p ∆set ApEp

p

where p= friction loss per unit length [kN/m] ∆set =anchorage set = 6mm

[5-14]

5.4.2 Creep and Shrinkage Losses

Creep and shrinkage losses for girders with external, unbonded tendons are computed using

Equation 5-15 which is modified from Menn’s formula for bonded tendons using new compatibility

conditions (Menn 1990).

∆P t =

nρ Acφ t 1lp σc0 x dx + εcs t EcAc

1 + nρ 1 + μφ t 1 +Ac

lpIc e2 x dx

[5-15]

where t = time measured from initial loading e(x) = distance from centroid of gross uncracked concrete section

to centroid of tendon Ec= modulus of elasticity of concrete at time of initial loading Ep= modulus of elasticity of prestressing tendon

n = Ep/Ec

Ac = area of gross uncracked concrete section Ap = area of tendon

Ic = moment of inertia of gross uncracked concrete section ϕ t = time-varying creep coefficient

= 1.5 for precast concrete = 2.0 for cast-in-place concrete

μ = aging coefficient = 0.8 (Menn 1990) εcs t = time-varying shrinkage strain under relative humidity of 60% Toronto value from CHBDC

σc0 x = concrete stress at tendon level due to initial load lp = arc length of tendon between anchors

Integrations are made over length of tendon from anchors to anchors

The creep and shrinkage losses are summarized in Table 5-3.

59

Table 5-3. Prestress losses due to creep and shrinkage

Creep and shrinkage losses (% of fPu)

L/h 30m 40m 50m

15 7.85% 6.24% 2.10%

20 6.30% 9.71% 6.11%

25 − 5.91% 4.35%

5.4.3 Losses due to Relaxation of Prestressing Steel

Prestress loss due to relaxation of steel (∆σp,rel) is estimated to be 59.3 MPa (3.19% fpu) for all

cases according to the following relationship suggested by Menn (1990).

Figure 5-12. Long-term loss of prestress due to relaxation (Menn 1990)

5.4.4 Total Prestress Losses

The effective prestress after all losses for every case is summarized in the following table. It is

the average value along the length of the tendon.

Table 5-4. Effective prestress after all losses

Effective prestress after all

losses (% of fPu)

L/h 30m 40m 50m

15 61% 62% 66%

20 64% 60% 63%

25 − 64% 66%

As shown in the table, compared to the estimated value of effective prestress (60% fPu), the gain

in stress is less than 6% after an explicit calculation. This gain in stress has an insignificant impact

on prestress requirements, so an effective prestress after all losses of 60% fpu is used in

serviceability checks.

0

2

4

6

8

10

0.5 0.55 0.6 0.65 0.7 0.75

∆σp,rel /σp0

[%]

σp0/fpu

60

5.5 Behaviour of Unbonded Tendons at Ultimate Limit States

The ULS flexural strength calculations for span-by-span method differ from those for previous

analyses. Previously, strain in bonded tendons can be directly solved with compatibility equations at

the specific section of interest. In contrast, span-by-span method uses unbonded tendons, the tendon

strain does not equal to the concrete strain at the same elevation and plane, and thus the tendon

stress cannot be determined locally. Furthermore, since strains in unbonded tendons are averaged

out between the anchorages, girders with unbonded tendons often have lower flexural resistances

compared to girders with bonded tendons. Figure 5-13 compares the behaviours of the two types of

tendons.

Figure 5-13. Compatibility conditions for bonded and unbonded tendons (Collins and Mitchell 1997)

To solve this problem, the unbonded tendon stress at ULS can be assumed to equal to the

effective prestress after all losses (60%fpu) instead of the yielding stress (90%fpu), because girders

with unbonded tendons often have lower flexural resistances than those with bonded tendons (Menn

1990). This conservative assumption causes the ULS be a governing factor for the prestress

requirement and results in extra tendons that are unnecessary for satisfying ULS limits.

In order to compute the actual prestress force in unbonded tendons at ULS, an iterative process

is used. This process computes the global deformation of the entire tendon instead of just the strains

61

at one plane. The iterative method is explained in greater details in Appendix B.4. The prestress at

ULS obtained from explicit calculations are summarized in the following table:

Table 5-5. Prestress at ULS

Prestress at ULS (% of fPu)

L/h 30m 40m 50m

15 73% 75% 69%

20 75% 76% 75%

25 − 82% 75%

These prestress at ULS are larger than the effective prestress after all losses, meaning that fewer

tendons are needed compared to the conservative method proposed previously.

5.6 Analysis Results

This section summarizes the analysis results including structural behaviours at ULS and SLS,

material consumptions to satisfy design requirements as well as the factors that limit the slenderness

of span-by-span precast box-girders.

5.6.1 Structural Behaviour and Dimensioning

The following table describes the ULS flexural strength and SLS stress at the most critical

location as well as the stirrup spacing requirement to satisfy ULS shear. The sizing of prestressing

tendons is discussed in Section 5.6.4. ULS flexural strength demand governs the prestress

requirement for all cases because unbonded tendons result in a lower tendon stress that is averaged

over the entire tendon length. Moreover, thermal gradient moments are not large enough to cause

overall positive moments at the piers, thus bottom tendons are not needed at these regions.

Table 5-6. Summary of structural response of precast span-by-span box-girder

L

(m)

L/h

Ultimate limit states Serviceability limit states

__Flexural strength__ __Shear strength__ __Stresses__ MULS (kNm) Mr (kNm) MULS/Mr Av (mm

2) smin (mm) % of girder @ smin σSLS (MPa) σSLS/fcr

30 15 38700 42900 90% 1200 207 15% 0.58 21%

20 38700 43300 89% 2000 247 4.5% -0.22 -

40

15 79400 83600 95% 1200 141 21% 1.81 67%

20 65500 68600 95% 1200 165 27% 1.99 71%

25 65400 71000 92% 1200 119 32% 1.95 70%

50

15 101000 102000 99% 1200 189 23% -0.10 -

20 97800 99400 98% 1200 149 29% 1.49 53%

25 97900 99700 98% 1200 116 33% 1.96 70%

5.6.2 Vibration Limits

As shown in the following graph, vibration limits are satisfied for all analysis cases and do not

affect tendon requirements. The truck deflections are at least 40% lower than the vibration

deflection limits.

62


5.6.3 Deflections

Deflection is not a limiting factor since the cross-sections are stiff and the resulting deflections

are insignificant. The maximum camber required is 93mm.


L=50m Limit

L=40m Limit

L=30m Limit

L=50m L=40m

L=30m

0

10

20

30

40

10 15 20 25 30

Truck loaddeflection

(mm)

L/h

L=30m

L=40m

L=50m

0

5000

10000

15000

20000

10 15 20 25 30

Dead load deflection

(L/∆)[down]

L/h

L=30m

L=40mL=50m

0

5000

10000

15000

10 15 20 25 30

Long-term deflection (creep + elastic) (L/∆)

[down]

L/h

L=30m

L=40m

L=50m

0

20000

40000

60000

10 15 20 25 30

Short-term deflection

(dead load+prestress

before loss) (L/∆) [up]

L/h

63

5.6.4 Material Consumption

The following table and graphs summarize the material consumption for each analysis case as

well as their variations from the baseline case (i.e. L/h=15). These results are also illustrated in

cross-section drawings in Appendix C.3. Like the cast-in-place box-girder case, as slenderness ratio

increases, concrete volume less compared to the increase in tendons. For instance, when the

concrete volume decreases by only 4% from the baseline case, the number of prestressing tendons

increases by 52% for the case with span of 30m and ratio of 20. On the other hand, reinforcing steel

mass remains similar for all the cases with the same span length.

Table 5-7. Summary of material consumption for precast span-by-span box-girder

L (m)

L/h


Volume (m

3)


Number of tendons


Mass (ton)


30 15 1920 0% 138 0% 159 0%

20 1840 -4.0% 210 +52% 149 -5.8%

40

15 2780 0% 198 0% 236 0%

20 2660 -4.4% 224 +13% 227 -3.9%

25 2600 -6.6% 296 +49% 231 -2.2%

50

15 3640 0% 204 0% 307 0%

20 3400 -6.7% 256 +25% 294 -4.2%

25 3350 -8.1% 333 +63% 299 -2.5%

a) Concrete volume

b) Number of prestressing strands

c) Reinforcing steel mass

Figure 5-16. Material consumptions for precast span-by-span box-girder

L=50m

L=40m

L=30m

0

1,000

2,000

3,000

4,000

10 15 20 25 30

Concretevolume

(m3)

L/h

L=50mL=40m

L=30m

0

100

200

300

400

10 15 20 25 30

Prestress strands

L/h

L=50m

L=40m

L=30m

0

100

200

300

400

10 15 20 25 30

Reinforcing steel mass

(ton)

L/h

64

5.6.5 Limiting Factors of Span-to-Depth Ratios

For all three span lengths analyzed (i.e. 30m, 40m, and 50m), the maximum slenderness ratio is

limited by the minimum height of access inside the box. As stated previously in Section 4.2.2.5, a

minimum access height of 1.0m is used to allow for sufficient space for construction, inspection and

maintenance. The maximum slenderness ratios are restricted to 20 and 25 for the cases with spans

of 30m and 40m respectively. For these two extreme cases, the access height is determined by the

height of the interior box cavity as demonstrated in Figure 5-17 which shows the cross-section for

the case with span of 40m and ratio of 25.

Figure 5-17. Access limited by height of interior box cavity

For the case with a span of 50m, the maximum span-to-depth ratio is 25 as shown in Figure 5-

18. Compared to the previous two cases, the access height is further reduced due to the use of

external tendons. If the slenderness ratio is increased to 30, the height of the interior box cavity and

the access height would reduce to 1.24m and 0.83m respectively.

Figure 5-18. Access limited by height of interior box cavity and external tendons

This minimum access height limit is not a rigid design requirement and is used merely to ensure

ease of construction. Higher span-to-depth ratios can be achieved for all three span lengths if only

ULS and SLS requirements are considered. However, as shown in Figures 5-17 and 5-18, the

addition prestress associated with higher slenderness ratios can only be accommodated with a

second layer of tendons. Such tendon layout is less efficient, because it reduces the tendon

eccentricity and as a result, more prestressing is needed. Due to the inefficient tendon configuration,

these slender cases are expected to be less cost-effective. Nonetheless, these cases are not

considered in this study since the minimum access requirement is not satisfied.

65

6 COST COMPARISONS

This cost study investigates the changes in superstructure and total construction costs, which are

based on the previously described material consumption results, when span-to-depth ratio varies.

Comparison of these costs reveals the optimal span-to-depth ratio for each bridge type. The cost

study also demonstrates the cost benefits of using the optimal ratios instead of conventional ratios.

Furthermore, a sensitivity analysis is performed to examine the impacts of changing material unit

prices and construction cost breakdown.

6.1 Material Costs

This section describes the material unit prices and provides a material cost comparison of all the

analysis cases.

6.1.1 Material Unit Prices

The unit costs for concrete, prestressing tendons and reinforcing steel are listed in Table 6-1.

They are based on values obtained from cast-in-place post-tensioned highway bridges in Ontario

(SNC-Lavalin 2008). All unit prices include installation costs (i.e. concrete placement and vibration,

grouting of tendons).

Table 6-1. Material unit prices

Material Unit Unit price

Concrete (f’c = 50 MPa) per m3 $ 1500

Material only per m3 $ 250

Formwork, falsework, and labour per m3 $ 1250

Longitudinal prestressing tendons (including anchorages) per kg $ 8.5

Reinforcing steel per kg $ 5.0

6.1.1.1 Concrete Material Unit Price

Concrete unit price varies with compressive strength, because higher strengths require more

cementing material and admixtures. Material unit prices for concrete with strengths from 10 MPa to

60 MPa are summarized in Table 6-2. For strengths greater than 60 MPa, values are extrapolated

from Figure 6-1 as indicated by hollow points. These prices account for the material only and do not

include grouting, pumping, formwork, falsework, and labour (Dufferin Concrete 2009).

66

Table 6-2. Concrete material unit price

Concrete compressive strengths (MPa)

Unit price (per m3)

10 $ 149 15 $ 154 20 $ 158 25 $ 164 30 $ 171 35 $ 180 40 $ 192 45 $ 220 50 $ 250 55 $ 262 60 $ 287 65 $ 320 70 $ 353 75 $ 389 80 $ 428

Figure 6-1. Concrete material unit price

6.1.1.2 Cast-in-Place versus Precast Concrete

The final results of this cost study are insensitive to the cost of casting and erection equipment

since the study does not compare the costs between different construction methods and cost

comparisons are mainly performed for bridges of the same type with varying spans and slenderness

ratios. Therefore, in the analysis, the unit price of precast concrete is assumed to be the same as the

one for cast-in-place concrete.

In a real construction setting, the casting operation would cost relatively the same for different

span lengths and construction methods, but the initial cost of precast is generally higher than cast-

in-place concrete, because it needs to be custom-built and stored in a manufacturing plant. Yet,

precast segments are more economically competitive when they are mass-produced. So, the unit

price of precast concrete depends on the length of the bridge. Precast concrete would likely cost

more if a short, 2-span bridge is analyzed instead of a long, 8-span viaduct. However, precast

concrete increases the speed of construction and reduces the risk of downtime created by having

additional on-site operations (e.g. in cast-in-place construction, workers need to build formworks,

install rebars, pour and vibrate concrete on-site). Precasting also eliminates other construction

problems related to cast-in-place such as poor quality control. These advantages of precast concrete

cannot be easily quantified, so they are not considered in this cost study.

As stated previously, the unit cost of precast concrete depends on the total length of the bridge

due to the relatively high initial cost associated with precasting. This dependency becomes less

significant when precast concrete are more widespread and the level of standardization increases.

The study anticipates that precast segments will be readily available like standard I-sections, thus

the impact of initial costs is reduced. As a result, the unit price of precast concrete is assumed to be

the same as cast-in-place concrete in the cost analysis.

Extrapolated values

$0

$100

$200

$300

$400

$500

0 20 40 60 80 100

Concrete material

price ($/m3)

Concrete strength (MPa)

67

6.1.1.3 Falsework versus Erection Truss

Span-by-span method has a cost reduction compared to cast-in-place on falsework since it does

not require falsework. Some cost gain is expected because a multiple-span viaduct allows for the

reuse of erection truss for each span while falsework needs to be assembled for the entire structure.

This cost advantage only exists for long bridges with repetitive spans. For shorter bridges with

fewer spans, the use of falsework might be more economical since the initial cost of erection truss is

relative high. Another factor that affects the cost of span-by-span bridges is the span length, because

the cost of erection equipment increases as span length increases. The cost of erection truss

becomes prohibitive when the span length is greater than 50m due to the heavy dead load (Hewson

2003). On the other hand, the unit price for falsework remains relatively constant as span length

increases. Therefore, a more extensive study is needed to investigate the relationship between cost,

span length and number of spans for cast-in-place on falsework and span-by-span methods, but this

is beyond the scope of this research.

6.1.1.4 Formwork

The unit price of formwork increases as the cross-section becomes more complex. For instance,

solid slabs require less complicated formworks compared to box-girders which have more surfaces.

Likewise, formworks for bridges with external tendons are simpler because the cross-section has no

internal tendon ducts. Variation in formwork price with respect to cross-sectional complexity is not

considered in the cost analysis and a constant concrete unit price is used for all cross-sectional types.

6.1.1.5 Prestressing Tendons

Both internal bonded and external unbonded tendons have the same unit price of $8.5/kg.

Although external unbonded tendons are more expensive in terms of material costs, they require

less labour due to the simple and rapid assembly. As a result, the cost is assumed to be the same as

internal bonded tendons.

6.1.2 Material Cost Comparisons

This section compares the material costs per deck area for all the analysis cases. Only the

superstructure material costs are compared because they are directly related to the span-to-depth

ratios whereas other components of construction costs (i.e. mobilization, substructure, and

accessories) are assumed to be independent of superstructure slenderness. The comparisons are

illustrated using a series of graphs in which the maximum cost variation for each span length is

indicated as a percentage next to the label for each line.

68

6.1.2.1 Concrete Cost Comparison

Figure 6-2 summarizes the pure material cost of concrete of all the analysis cases evaluated. In

general concrete cost decreases with increasing slenderness ratio. For solid slabs, however, cost

increases drastically for the very slender cases which require concrete strengths higher than 50 MPa.

Figure 6-2. Concrete material cost comparison

Figure 6-3. Total concrete cost comparison

L=20m,42%

L=25m, 12%

L=30m, 43%L=35m, 29%

L=35m, 24%

L=50m, 33%

L=60m, 40%

L=75m, 42%

L=30m, 4.2%L=40m, 7.0%L=50m, 8.8%

$0

$50

$100

$150

$200

$250

$300

5 10 15 20 25 30 35 40 45 50 55

Cost of concrete (material only)

per deckarea ($/m2)

L/h

L=20m

L=35m

L=30m

CIP on falsework box-girder

CIP on falsework solid slab

Precast span-by-span box-girder

L=20m, 11%

L=25m, 20%

L=30m, 24%

L=35m, 22%

L=35m, 24%

L=50m, 33%

L=60m, 40%

L=75m, 42%

L=30m, 4.2%L=40m, 7.0%L=50m, 8.8%

$0

$200

$400

$600

$800

$1,000

$1,200

$1,400

$1,600

5 10 15 20 25 30 35 40 45 50 55

Cost of concrete per deck

area ($/m2)

L/h

L=20m

L=35m

L=30m




69

The cost effect of increasing concrete strength diminishes when the costs of formwork and

falsework or erection truss are included as shown in Figure 6-3. In general, concrete cost decreases

as span-to-depth ratio increases due to the reduction in superstructure depth. However, decrease in

concrete cost slows down at higher span-to-depth ratios, because the cost reduction is offset by the

increase in bottom slab thickness to accommodate compressive stress regions in box-girders or by

the increase in concrete strengths to satisfy maximum reinforcement criterion in solid slabs.

Concrete cost also varies substantially between different cross-section types even when the

same span length and span-to-depth ratio are used. For instance, for cases with a span of 35m, the

concrete costs are $1500/m2 for a solid slab with span-to-depth ratio of 30 and $836/m

2 for a cast-

in-place box-girder with ratio of 25. Concrete cost for the thinner solid slab (depth = 1.2m) is 79%

more than the one for the deeper box-girder (depth = 1.4m) even though the cross-sectional depths

vary by only 17%. This comparison indicates a significant reduction in concrete consumption,

which translates into dead load reduction, occurs when a box section is used instead of a solid slab.

6.1.2.2 Prestressing Cost Comparison

Unlike concrete, the cost of prestressing tendons varies substantially as span-to-depth ratio rises

as shown in Figure 6-4.

Figure 6-4. Prestressing tendon cost comparison

The maximum cost differences amongst cases with the same span length are 284% for cast-in-

place on falsework box-girder, 60% for cast-in-place on falsework solid slab, and 62% for precast

L=20m, 53%

L=25m, 40%

L=30m, 60%

L=35m, 39%

L=35m, 161%

L=50m, 209%

L=60m, 223%

L=75m, 284% L=30m, 53%

L=40m, 49%

L=50m, 62%

$0

$50

$100

$150

$200

$250

$300

$350

$400

$450

$500

5 10 15 20 25 30 35 40 45 50 55

Cost of prestress steel per deck

area ($/m2)

L/h

L=20m

L=35m

L=30m




70

span-by-span box-girder. The increases in prestressing demand are significant because although the

dead load (i.e. concrete volume) decreases as girder becomes more slender, this load reduction

cannot compensate for the relatively large decline in moment resistance which is attributed to the

decrease in flexural stiffness of the cross-section and in efficiency of tendon layout (tendon

eccentricity is lower for slender sections). This problem is illustrated in Table 6-3 which shows the

maximum change in concrete volume, girder depth, moment of inertia (proportional to flexural

stiffness), and prestress demand between the case with the lowest span-to-depth ratio and the case

with the highest ratio for a particular span length for each bridge type. For all three bridge types, the

decrease in dead load is small relative to the decrease in girder depth and modulus of elasticity. As a

result, more prestressing steels are needed to offset the reduced load capacity.

Table 6-3. Comparison of changes in cross-sectional depth and prestressing demand

Factor CIP on falsework box-girder (L=75m)

CIP on falsework solid slab (L=30m)

Precast span-by-span box-girder (L=50m)

Concrete volume -29% -26% -8.1% Girder depth -71% -33% -40% Moment of inertia -94% -70% -69% Prestress demand +284% +60% +62%

A comparison of prestressing steel costs between bridge types confirms that solid slab is the

least efficient in resisting loads. For example, for a span length of 35m, the prestressing tendons

required for a solid slab with a span-to-depth ratio of 30 is $275/m2 which is 2.4 times greater than

the $117/m2 needed for a cast-in-place box-girder with span-to-depth ratio of 25. Solid slabs require

more prestressing per unit deck area because they are heavier than box-girders with the same span

lengths and depths. Precast span-by-span box-girder is more efficient than solid slabs but less

efficient than cast-in-place box-girder, because it uses external tendons which reduce tendon

eccentricities and thus lowers the prestress moments. For instance, for the cases with span length of

50m and span-to-depth ratio of 25, the prestressing tendons required in a precast box-girder

($215/m2) cost 30% more than the tendons needed in a cast-in-place box-girder ($165/m

2) although

the precast case has 7.6% less dead load.

6.1.2.3 Reinforcing Steel Cost Comparison

Cost of non-prestressed reinforcing steel depends on the shear stirrup requirement and concrete

volume. Figure 6-5 shows the cost of stirrups in solid lines and the cost of minimum reinforcing

steels in dashed lines. In this study, the minimum reinforcing steel is defined as all the non-

prestressed steels other than stirrups and is illustrated in Figures 4-3, 4-15 and 5-4.

71

Figure 6-5. Cost comparison of stirrups and minimum reinforcing steel

As shown in Figure 6-5, cost of stirrups generally increases with span-to-depth ratios since

more stirrups are needed for slender cases which have lower shear resistances. The increase in

stirrup cost for cast-in-place box-girders slows down for ratios beyond 30, because 25M bars, which

result in a more efficient stirrup layout for these cases, is used instead of 20M bars. Also, solid slabs

require fewer stirrups compared to box-girders because they have longer effective widths and

higher concrete strengths for slender cases which result in greater concrete shear resistances. The

large number of prestressing tendons, which are needed for adequate flexural resistances, further

helps in resisting shears. For instance, for the cases with a span length of 35m, the stirrups required

for a solid slab with a span-to-depth ratio of 30 cost $15/m2 while they cost 3.2 times more for a

box-girder with a ratio of 25 (i.e. $48/m2).

The dashed lines on Figure 6-5 show the costs of minimum reinforcing steel required for

stability of the steel bar cage and for crack control. The minimum requirement of reinforcing steel is

proportional to the concrete volume, so it decreases as span-to-depth ratio increases.

The total cost of reinforcing steel is the sum of the costs of stirrups and minimum reinforcing

steel and the contributions of these two cost components vary over different span-to-depth ratios.

Figure 6-6 illustrates the contribution of each component by comparing the percentage distributions

and the actual costs for the longest span case for each bridge type. As shown in Figure 6-6 a, the

impact of stirrups on the total reinforcing steel cost increases as the girder becomes more slender

L=20m, 423%L=25m, 180%

L=30m, 413%L=35m, 257%

L=35m, 40%L=50m, 76%L=60m, 84%L=75m, 110%

L=30m, 18%

L=40m, 18%L=50m, 23%

L=20m, 22%L=25m, 23% L=30m, 35%L=35m, 37%

L=35m, 24%

L=50m, 33%

L=60m, 40%

L=75m, 42%

L=30m, 4.2%L=40m, 7.0%L=50m, 8.8%

$0

$50

$100

$150

$200

$250

$300

$350

$400

5 10 15 20 25 30 35 40 45 50 55

Cost of reinforcing steel per deck area

($/m2)

L/h

L=20m

L=35m

L=30m

L=50m

L=30m



Precast span-by-span box-girderStirrups

Minimum reinforcing steel

72

regardless of bridge type. This trend is a result of the reduction in minimum reinforcement due to

decreasing concrete volume and the rise in shear demand due to increasing girder slenderness.

As shown in Figure 6-6 b, for both cast-in-place and precast box-girders, the total reinforcing

steel cost decreases as span-to-depth ratio increases until 20. For these cases, decrease in minimum

reinforcing steel is significant due to the large reduction in concrete volume (shown in Figures 4-8 a

and 5-16 a) while the increase in stirrups is relatively small. As a result, the total reinforcing steel

cost exhibits an overall decreasing trend. When span-to-depth ratio increases beyond 20, the

reduction in reinforcing steel diminishes and becomes insignificant compared to the increase in

stirrups, resulting in an overall increase in total reinforcing steel cost.

For solid slabs, the total reinforcing steel cost does not increase significantly even for span-to-

depth ratios above 20. Although the stirrups cost increases by more than two times over the entire

range of ratios, this increase is offset by the equally large decrease in minimum reinforcement cost.

The decrease in minimum reinforcement is relatively large compare to the one for box-girders with

the same slenderness ratios, because solid slabs experience constant volume reduction as span-to-

depth ratio increases whereas the volume diminishes at a decreasing rate for box-girders (discussed

previously in Section 4.3.3.5 and illustrated in Figure 4-20 a).

a) Percentage distribution of reinforcing steel cost

b) Distribution of reinforcing steel cost

Figure 6-6. Cost distribution of stirrups and minimum reinforcing steel

0%

20%

40%

60%

80%

100%

10 15 20 25 30 35 30 35 40 45 15 20 25

Minimum reinforcing steelStirrups

$0

$100

$200

$300

$400

10 15 20 25 30 35 30 35 40 45 15 20 25

Minimum reinforcing steel

Stirrups

L=75mCIP on falsework box-girder

L=35mCIP on falsework solid slab

L=50mPrecast span-by-span box-girder

73

The total cost of reinforcing steel for all analysis cases is shown in Figure 6-7. The maximum

variations in total reinforcing steel cost over the analysis range of ratios are only 18% for cast-in-

place on falsework box-girder, 5.8% for cast-in-place on falsework solid slab, and 6.2% for precast

span-by-span box-girder.

Figure 6-7. Total reinforcing steel cost comparison

6.1.2.4 Total Superstructure Cost

Figure 6-8 describes the total superstructure material cost, which consists of costs of concrete

(material only), prestressing tendon, and reinforcing steel. The contribution of these three

components are summarized in Figure 6-8 a. For deep box-girders, the superstructure material cost

is mainly governed by concrete and reinforcing steel costs which decrease as span-to-depth ratio

increases due to the reduction in concrete volume. The effect of increasing prestressing tendon cost

is relatively minimal and is overshadowed by the reduction in concrete and reinforcement costs.

Therefore, the total material cost exhibits an overall decreasing trend for low span-to-depth ratios.

For the slender box-girders, the costs of prestressing tendon and reinforcing steel rise as span-to-

depth ratio increases. This cost increase is greater than the concrete cost reduction since the

decreasing in concrete volume slows down at higher span-to-depth ratios. As a result, the total

material cost experiences a net increase at higher ratios. For slender solid slabs, increase in material

cost is also attributed to the increase in concrete cost from the use of higher strength concrete.

L=20m, 2.7%L=25m, 2.5%

L=30m, 5.8%L=35m, 5.4%

L=35m, 17%

L=50m, 18%

L=60m, 18%

L=75m, 18%

L=30m, 6.2%

L=40m, 4.0%L=50m, 4.3%

$0

$50

$100

$150

$200

$250

$300

$350

$400

5 10 15 20 25 30 35 40 45 50 55

Cost of reinforcing

steel per deck area

($/m2)

L/h

L=20m

L=35m

L=30m




74

a) Percentage distribution of superstructure material cost

b) Total superstructure material cost

Figure 6-8. Total superstructure material cost comparison

Figure 6-9 describes the total superstructure cost which includes the falsework and casting costs

of concrete in addition to the total material cost. As stated in Section 6.1.1, the cost of concrete

casting and falsework is a function of concrete volume and is not related to concrete strength, and

thus it generally decreases as span-to-depth ratio increases. This decreasing casting and falsework

cost amplifies the cost reduction for deep girders and dampens the cost increase for slender girders.

The dampening effect is especially obvious for solid slabs in which the concrete volume reduction

associated with rising span-to-depth ratio is significant.

0%

20%

40%

60%

80%

100%

10 15 20 25 30 35 30 35 40 45 15 20 25

Prestressing tendon

Reinforcing steel

Concrete (material only)




L=20m, 28%L=25m, 11%

L=30m, 28%

L=35m, 27%

L=35m, 6.3%L=50m, 19%L=60m, 22%

L=75m, 42%

L=30m, 6.7%

L=40m, 9.2%L=50m, 12%

$0

$500

$1,000

$1,500

$2,000

5 10 15 20 25 30 35 40 45 50 55

Total material cost per deck area ($/m2)

L/h

L=20m

L=35m

L=30m




75

a) Percentage distribution of superstructure cost

b) Total superstructure cost

Figure 6-9. Total superstructure cost comparison (including cost of concrete placement)

The optimal span-to-depth ratio in terms of superstructure costs is determined by balancing the

concrete cost reduction with the prestressing cost increase as span-to-depth ratio rises. In terms of

superstructure material costs only, the cost optimal ratios are 15, 30, and 15 for cast-in-place on

falsework box-girder, solid slab and precast span-by-span box-girder respectively. These optimal

ratios increase to 25, 40, and 20 if the total superstructure costs are considered. This improvement

in slenderness ratio demonstrates the construction economy related to slender and lighter

superstructures.

Although these optimal ratios based on total superstructure costs are higher than the

conventional ones, the changes in superstructure cost between these two sets of ratios are actually

0%

20%

40%

60%

80%

100%

10 15 20 25 30 35 30 35 40 45 15 20 25

Prestressing tendon

Reinforcing steel

Concrete (material and placement)




L=20m, 3.6%

L=25m, 9.6%

L=30m, 12%

L=35m, 12%

L=35m, 14%

L=50m, 17%

L=60m, 20%

L=75m, 18%

L=30m, 0.3%

L=40m, 2.6%L=50m, 3.3%

$0

$500

$1,000

$1,500

$2,000

5 10 15 20 25 30 35 40 45 50 55

Superstructurecost per

deck area ($/m2)

L/h

L=20m

L=35m

L=30m




76

minor (less than 11%) as shown in Table 6-4. Using the more cost-efficient ratios instead of

conventional ratios yields only insignificant total superstructure cost savings compared to the

changes in individual material costs. The table also shows the maximum cost variations within the

entire range of slenderness ratios analyzed in parentheses. Even if the least cost-efficient ratio is

used, the total superstructure cost increases by only 20%.

Table 6-4. Total superstructure cost variations

CIP on falsework box-girder CIP on falsework solid slab Precast span-by-span box-girder

Analysis range of ratios 10 - 35 30 - 50 15 - 30 Conventional ratio 20 30 17 Optimal ratio 25 40 20

Cost component Concrete -5.1% (42%) -19% (24%) -4.1% (8.8%) Prestressing tendon +28% (284%) +53% (60%) +26% (62%) Reinforcing steel -1.3% (18%) -5.3% (5.8%) -3.6% (6.2%) Total superstructure -0.6% (20%) -11% (12%) -1.8% (3.3%)

Furthermore, Figure 6-9 shows that instead of refining cross-sectional component dimensions,

using a more efficient structural system results in greater cost savings. For instance, for a span

length of 35m, a cast-in-place solid slab with a slenderness ratio of 30 costs 59% more than a cast-

in-place box-girder with a ratio of 25. Therefore, the superstructure cost savings associated with

optimizing the span-to-depth ratios are negligible compared to the savings from choosing the proper

bridge type for a given span length and girder depth.

6.2 Overall Construction Costs

This section discusses the impact of optimizing span-to-depth ratio on the total construction cost

which includes the costs of superstructure, substructure, mobilization and accessories such as

bearings. First, the percentage breakdown of these cost components with respect to the total

construction cost is described in Section 6.2.1. Section 6.2.2 then describes the total construction

costs for all analysis cases computed using this cost breakdown. The analysis results indicate that

varying the span-to-depth ratio has minor influence on the total construction cost, because only the

superstructure cost depends on span-to-depth ratio whereas the cost of substructure and accessories

are related more to the span length.

6.2.1 Construction Cost Breakdown

The construction cost breakdown used in the analysis is adopted from Menn’s cost study on 19

concrete highway bridges constructed between 1958 and 1985 in Switzerland (Table 6-5).

Mobilization comprises of all the work that needs to be completed prior to the onset of construction.

Such tasks include preparing site facilities, procuring equipments and establishing access to the

construction site. Some examples of accessories are bearings, expansion joints, and drainage

77

systems. These two factors are independent of span-to-depth ratio, but they are related to the bridge

site conditions, number of spans, and bridge length. Likewise, the substructure cost depends more

on geotechnical conditions and bridge height. Therefore, the costs for these three items are

computed for the baseline cases only and are assumed to be constant for all other analysis cases

with the same span length.

Table 6-5. Construction cost breakdown (Menn 1990)

Item

Cost (% of total construction cost)

Mobilization 8.0%

Structure

Substructure

Foundations 18.0%

Piers and abutments 5.5%

Total substructure 23.5% 23.5%

Superstructure

Total superstructure 54.5%

Total structure 78.0% 78.0%

Accessories 14.0%

Total construction cost 100.0%

6.2.2 Total Construction Cost Comparison

The total construction costs for all analysis cases are illustrated in Figure 6-10 and a sample

calculation is shown in Appendix B.5. The trend and the most cost-optimal ratios are the same as

the ones for superstructure cost, because the total construction cost is simply the sum of

superstructure cost and the costs of substructure, mobilization and accessories which are constant

amongst analysis cases with the same span length. However, factoring in this constant cost further

diminishes the economic incentive of optimizing span-to-depth ratios. In fact, the total construction

cost premiums of using the optimal ratios instead of conventional ratios are only 0.4% for cast-in-

place box-girder, 5.8% for cast-in-place solid slab, and 1.0% for precast span-by-span box-girder.

Even within the entire feasible range of span-to-depth ratios, maximum variations in total

construction cost are only 11%, 6.2% and 1.8% for the three bridge types respectively.

These cost reductions related to optimizing span-to-depth ratios for a given span are

insignificant compared to the cost reduction from choosing a suitable bridge type and span

arrangement. For instance, for a span length of 35m, using a cast-in-place box-girder with

slenderness ratio of 25 instead of a cast-in-place solid slab with a ratio of 30 reduces the total

construction cost by 37%. Cost improvement can also be attained by using shorter span lengths. For

instance, using a span length of 50m instead of 75m reduces the total construction cost by 20% in

cast-in-place box-girders.

78

Figure 6-10. Total construction cost comparison

6.3 Other Cost Factors

In addition to construction costs, bridge designers are also concerned with other cost factors

such as operation, maintenance, rehabilitation, and demolition costs. These factors are excluded in

this cost study, but changing span-to-depth ratio is expected to have minor effects on these costs.

These factors depend more on the overall design concept (e.g. structural system, cross-section type,

span arrangement) than on the dimensioning of structural components (Menn 1990).

L=20m, 1.9%

L=25m, 5.0%

L=30m, 6.2%

L=35m, 6.1%

L=35m, 7.7%

L=50m, 9.0%

L=60m, 11%

L=75m, 10%

L=30m, 0.1%

L=40m, 1.4%L=50m, 1.8%

$0

$500

$1,000

$1,500

$2,000

$2,500

$3,000

$3,500

$4,000

5 10 15 20 25 30 35 40 45 50 55

Total construction

cost perdeck area

($/m2)

L/h

L=20m

L=35m

L=30m




79

6.4 Sensitivity Analysis

The purpose of the sensitivity study is to investigate how sensitive are the analysis results to

changes in material unit prices and in breakdown of construction costs. The material unit prices

vary in different locations due to differences in labour costs and technology levels. The construction

cost breakdown also changes depending on the site conditions.

6.4.1 Sensitivity with Respect to Changes in Material Unit Prices

The first aspect to investigate is the impact of changes in material unit prices. A new set of

superstructure costs are calculated based on the following variations in material unit prices:

Table 6-6. Material unit price changes

Material Unit Original price +50% -50%

Concrete per m3

$ 1,500 $ 2,250 $ 750

Prestressing tendon per kg $ 8.50 $ 12.75 $ 4.25

Reinforcing steel per kg $ 5.00 $ 7.50 $ 2.50

These changes in unit prices are drastic and are chosen for only illustrative purposes such that a

clear trend would be observed. A more realistic maximum price change would be around 20%.

The effects of changing unit prices on total construction costs are illustrated in the following

graphs. The maximum percentage differences in cost for each span length over the specified range

of span-to-depth ratios are also indicated on the graphs. The dashed lines represent the costs under

original unit prices while the solid lines represent the costs when the unit prices vary by 50%. The

impacts of altering unit prices are illustrated in Figures 6-11 and 6-12 for concrete, Figures 6-13 and

6-14 for prestressing tendons, and Figures 6-15 and 6-16 for reinforcing steel. The graphs show

that modifying concrete unit price has the greatest influence on the total construction cost, because

concrete cost constitutes a large portion of the superstructure cost. For example, for a cast-in-place

on falsework box-girder with span length of 75m and span-to-depth ratio of 10, increasing the

concrete unit cost by 50% raises the total construction cost by 35%. On the other hand, increasing

the prestressing cost by 50% only increases the total construction cost by 4.6%.

80

Figure 6-11. Total construction cost comparison (+50% concrete unit price)

Figure 6-12. Total construction cost comparison (-50% concrete unit price)

L=20m, 3.8%

L=25m, 6.4%

L=30m, 8.0%

L=35m, 8.7%

L=35m, 9.1%

L=50m, 11%

L=60m, 13%

L=75m, 12%

L=30m, 0.5%

L=40m, 1.7%L=50m, 2.2%

$0

$1,000

$2,000

$3,000

$4,000

$5,000

0 5 10 15 20 25 30 35 40 45 50 55

Total construction

cost perdeck area

($/m2)

L/h

L=35m

L=75m

L=50m




L=20m, 2.8%L=25m, 1.9%

L=30m, 3.1%

L=35m, 3.8%

L=35m, 5.4%L=50m, 6.6% L=60m, 7.8%

L=75m, 12%

L=30m, 1.5%

L=40m, 2.8%

L=50m, 3.1%

$0

$1,000

$2,000

$3,000

$4,000

$5,000

0 5 10 15 20 25 30 35 40 45 50 55

Total construction

cost perdeck area

($/m2)

L/h

L=25m

L=75m

L=50m




81

Figure 6-13. Total construction cost comparison (+50% prestressing tendon unit price)

Figure 6-14. Total construction cost comparison (-50% prestressing tendon unit price)

L=20m, 2.4%

L=25m, 3.5%

L=30m, 4.9%

L=35m, 5.5%

L=35m, 6.0%

L=50m, 7.4%

L=60m, 8.8%

L=75m, 11%

L=30m, 1.3%

L=40m, 2.3%

L=50m, 2.7%

$0

$1,000

$2,000

$3,000

$4,000

$5,000

0 5 10 15 20 25 30 35 40 45 50 55

Total construction

cost perdeck area

($/m2)

L/h

L=25m

L=75m

L=50m




L=20m, 3.9%

L=25m, 6.8%

L=30m, 8.3%

L=35m, 9.3%

L=35m, 9.7%

L=50m, 12%L=60m, 13%

L=75m, 13%

L=30m, 1.1%L=40m, 1.9%L=50m, 2.5%

$0

$1,000

$2,000

$3,000

$4,000

$5,000

0 5 10 15 20 25 30 35 40 45 50 55

Total construction

cost perdeck area

($/m2)

L/h

L=35m

L=75m

L=50m




82

Figure 6-15. Total construction cost comparison (+50% reinforcing steel unit price)

Figure 6-16. Total construction cost comparison (-50% reinforcing steel unit price)

As shown in the graphs, the optimal span-to-depth ratios remain at 25, 40, and 20 for cast-in-

place box-girder, cast-in-place solid slab, and precast box-girder regardless of unit price changes.

Table 6-7 summarizes the maximum cost improvements of using these optimal span-to-depth ratios

L=20m, 2.9%

L=25m, 4.7%

L=30m, 6.1%

L=35m, 6.7%

L=35m, 7.7%

L=50m, 9.1%

L=60m, 11%

L=75m, 10%

L=30m, 0.2%

L=40m, 1.5%L=50m, 1.7%

$0

$1,000

$2,000

$3,000

$4,000

$5,000

0 5 10 15 20 25 30 35 40 45 50 55

Total construction

cost perdeck area

($/m2)

L/h

L=35m

L=75m

L=50m




L=20m, 3.2%

L=25m, 5.3%

L=30m, 6.9%

L=35m, 7.4%

L=35m, 7.6%

L=50m, 9.0%L=60m, 11%

L=75m, 10%

L=30m, 0.5%

L=40m, 1.5%L=50m, 1.8%

$0

$1,000

$2,000

$3,000

$4,000

$5,000

0 5 10 15 20 25 30 35 40 45 50 55

Total construction

cost perdeck area

($/m2)

L/h

L=35m

L=75m

L=50m




83

instead of conventional ratios for the specified variations in unit price. Despite such drastic unit

price changes, the cost improvements are only 0.6% ± 1.1% for cast-in-place box-girder, 5.8% ±

2.3% for cast-in-place solid-slabs, and 1.0% ± 0.5% for precast span-by-span box-girder.

Table 6-7 also lists the maximum percentage changes in cost over the entire range of span-to-

depth ratio in parentheses. These values represent the maximum cost increase when the least cost-

efficient span-to-depth ratios are used instead of the optimal ratios. For instance, for a given span

length, using a span-to-depth ratio of 10 instead of the optimal ratio results in 11% increase in total

construction cost for cast-in-place box-girder. This increase in cost changes to 12% if the concrete

unit price is increased by 50%. For cases with altered unit prices, the changes in cost with respect to

span-to-depth ratios deviate less than 3% from the ones for cases with original unit prices. Since the

optimal ratios remain the same and the changes in cost improvements are negligible (less than 3%),

results from the cost study are insensitive to modifications in unit prices.

Table 6-7. Summary of material unit price sensitivity analysis

Unit cost change CIP on falsework box-girder CIP on falsework solid slab Precast span-by-span box-girder

Range of L/h ratio 10 - 35 30 - 50 15 - 30

Optimal ratio 25 40 20

Conventional ratio 20 30 17

Original unit prices 0.4% (11%) 5.8% (6.2%) 1.0% (1.8%)

+50% concrete 0.9% (12%) 7.9% (8.7%) 1.3% (2.2%)

-50% concrete 1.5% (12%) 3.6% (3.8%) 0.5% (3.1%)

+50% prestressing tendon 0.9% (11%) 5.2% (5.5%) 0.8% (2.7%)

-50% prestressing tendon 1.0% (13%) 8.1% (9.3%) 1.5% (2.5%)

+50% reinforcing steel 0.3% (11%) 6.3% (6.7%) 1.0% (1.7%)

-50% reinforcing steel 0.4% (11%) 6.9% (7.4%) 0.9% (1.8%)

6.4.2 Sensitivity with Respect to Changes in Construction Cost Breakdown

The purpose of this section of the sensitivity study is to investigate the impact of altering the

construction cost breakdown. Hitherto, the cost results are computed according to Menn’s

breakdown of construction costs as described in Section 6.2.1. This breakdown changes under

situations such as complications in geotechnical and hydraulic conditions, changes in bridge height

which affect pier costs, and variations in mobilization cost for different construction sites.

Previous cost calculations assume that the superstructure costs constitute 54.5% of the total

construction cost for the baseline cases. Costs for the remaining items (i.e. substructure,

mobilization and accessories) are computed based on the superstructure costs of the baseline cases

and are set to be constant for the other cases with the same span lengths. Therefore, altering only the

proportion of superstructure cost is sufficient in demonstrating the influence of different

construction cost breakdowns. The total construction costs obtained from varying the superstructure

component from 20% to 80% for cast-in-place on falsework box-girder with a span length of 50m

are plotted in Figure 6-17.

84

As shown in the graph, changing the superstructure percentage basically shifts the curves

vertically while the cost difference between cases with different span-to-depth ratios remains the

same for each curve. As a result, the optimal ratio remains at 25 regardless of changes in the cost

breakdown. The graph also shows the percentage cost improvements from conventional ratio to

optimal ratio on the left (maximum cost variations over the entire range of span-to-depth ratios are

shown in parentheses). Obviously, the cost improvement increases as superstructure percentage

increases because the effect of reducing superstructure cost on the total construction cost is greater

if superstructure cost forms a large portion of the total cost. Therefore, using optimal span-to-depth

ratios poses more economic incentive if the superstructure percentage is higher. However, in spite

of the changes in the cost breakdown, the saving from using the optimal ratio instead of

conventional ratio is still less than 0.5% while the maximum cost variation within the entire range

of ratio is less than 13%. The same pattern occurs in the other bridge types and span lengths as

shown in Appendix C.4. Therefore, the values for optimal span-to-depth ratio as well as the cost

variations between cases with optimal and conventional ratios are insensitive to changes in cost

breakdown.

Figure 6-17. Total construction costs under changes in construction cost breakdown

0.3% (6.9%)40%

0.3% (8.6%)50%0.4% (9.0%)54.5% (Menn)0.4% (10%)60%0.5% (12%)70%0.5% (13%)80%

0.1% (3.4%)

Cost variation Superstructure as % of total construction cost

20%

0.2% (5.2%)30%

$0

$2,000

$4,000

$6,000

$8,000

0 5 10 15 20 25 30 35

Total construction cost per deck area

($/m2)

L/h

85


The results of this cost study are summarized in Table 6-8 which shows the percentage changes

in cost when optimal ratios instead of the conventional ones are used. Cost variations over the

analysis range of ratios are also included within the parentheses. For all three bridge types, the cost-

effective ratios are higher than the conventional ratios, but the actual cost saving associated with

using optimal ratios is less than 5.8%. The cost study also determines that the maximum cost

variability is less than 11%, meaning that using the least cost-efficient ratio within the analysis

range would only incur a relatively small additional cost. The range of cost-optimal ratios is thus

expanded from the typical ranges defined in Chapter 2 to the analysis ranges of ratios indicated in

Table 6-8.

Table 6-8. Summary of cost study

The results in Table 6-8 are found to be insensitive to changes in material unit price and

construction cost breakdown. The optimal ratios, however, are determined based on parameters

defined specifically for this study such as cross-section dimensions and span arrangements. If these

parameters are altered, the optimums would likely be different. For example, the optimal ratios

might increase if thicker webs are used for box-girders, because the prestressing requirement is

reduced for slender cases due to the more efficient tendon layout as discussed in Section 4.2.2.5.

Therefore, the actual values of optimal ratios determined in this study are expected to change in a

real situation.

Yet, the general finding regarding the variability in cost is still valid for a broad range of

situations, because the study demonstrates that cost savings from the use of optimal ratios are minor

compared to other construction cost components which are independent of span-to-depth ratio (e.g.

costs of mobilization, substructure, and accessories). So, even if optimal ratio changes, the

variability in cost is expected to remain relatively insignificant over the analysis range of ratios.

CIP on falsework box-girder CIP on falsework solid slab Precast span-by-span box-girder

Analysis range of ratios 10 - 35 30 - 50 15 - 30 Typical range of ratios 17.7 - 22.6 22 - 39 15.7 - 18.8 Conventional ratio 20 30 17 Cost-optimal ratio 25 40 20

Cost component Concrete -5.1% (42%) -19% (24%) -4.1% (8.8%) Prestressing tendon +28% (284%) +53% (60%) +26% (62%) Reinforcing steel -1.3% (18%) -5.3% (5.8%) -3.6% (6.2%) Total superstructure -0.6% (20%) -11% (12%) -1.8% (3.3%) Total construction cost -0.4% (11%) -5.8% (6.2%) -1.0% (1.8%)

86

7 AESTHETICS COMPARISONS

The selection of slenderness ratio has significant impact on the overall appearance of girder

bridges. In particular, the ratio is an especially important visual criterion for highway overpasses,

which are mostly observed from the highways passing beneath them, because the superstructure is

the “prime object of scrutiny” from this view point (Elliott 1991). The previous cost study

demonstrates that total construction cost varies by less than 11% over the range of span-to-depth

ratios investigated. This finding provides the designer with more freedom for aesthetic expressions

since he can choose from a wide range of slenderness ratios without much economic restrictions.

This chapter examines these aesthetic opportunities by comparing the visual impacts of different

span-to-depth ratios. This chapter also discusses some visually superior slenderness ratios by

exploring existing bridges, which have been considered by the general public as aesthetically

pleasing, and by examining some past studies on bridge aesthetics.

7.1 Visual Impact of Span-to-Depth Ratio

This section demonstrates the visual impact of altering slenderness ratios by comparing

drawings of bridges with various ratios as shown in Figures 7-1 to 7-3. These figures compare

bridges with: a) conventional ratio obtained from Chapter 2; b) optimal ratio in terms of cost

efficiency determined in Chapter 6; c) least cost-efficient ratio for each of the three bridge types

considered. The total construction cost and its percentage variation from the cost of the optimal

ratio are indicated in the parentheses. Each set of drawing includes a 3-D rendering from the

vantage point of a driver, who is passing under the overpass at 150m away from the bridge, as well

as 2-D elevation and cross-sectional views. The 2-D drawings are included to illustrate that

although changing the slenderness ratio results in clear visual differences on paper, such differences

might not be as apparent from the driver’s perspective in a real situation. The drawings do not

include barriers; the effect of barriers is discussed in Section 7.1.2. Also, all the 3-D drawings are

obtained under the same lighting condition.

87

a) Conventional L/h = 20 ($2450/m2, 0.4%)

b) Optimal L/h = 25 ($2460/m2)

c) Maximum cost L/h = 10 ($2670/m2, 9.0%)

Figure 7-1. Cast-in-place on falsework box-girder with L=50m

88

a) & c) Conventional and maximum cost L/h = 30 ($3000/m2, 5.8%)

b) Optimal L/h = 40 ($2830/m2)

Figure 7-2. Cast-in-place on falsework solid slab with L=30m

89

a) Conventional L/h = 17 ($2370/m2, 1.0%)

b) Optimal L/h = 20 ($2350/m2)

c) Maximum cost L/h = 25 ($2390/m2, 1.8%)

Figure 7-3. Precast segmental span-by-span box-girder with L=50m

As shown in Figures 7-1 and 7-3, using cost-effective span-to-depth ratios noticeably improves

visual slenderness compared to the conventional ratios for cast-in-place on falsework and precast

span-by-span box-girders. The visual difference is even more apparent when the optimal case is

compared to the case with maximum construction cost. This finding indicates that varying the span-

to-depth ratio can have a significant visual impact without substantial cost premiums (less than 11%

variation in cost for all the analysis cases considered).

90

However, the visual impact of changing span-to-depth ratio is less obvious for the solid slab

case, because visual difference diminishes as slenderness ratio increases beyond 25 as shown in

Figure 7-4.

a) L/h=10

b) L/h=15

c) L/h=20

d) L/h=25

e) L/h=30

f) L/h=35

Figure 7-4. Visual effects of increasing span-to-depth ratios from 10 to 35

91

As stated previously, increasing the span-to-depth ratio by reducing the girder depth can

enhance visual slenderness without incurring significant additional costs. Increasing the ratio by

extending the span length has similar visual effect as shown in Figure 7-5. Yet, the cost premium

associated with increasing span length is much more severe. For instance, for the cast-in-place box-

girder case depicted in the figure, increasing the span length from 50m to 75m adds 23% to the total

construction cost.

a) L/h=20, L=50m ($2450/m2)

b) L/h=30, L=75m ($3020/m2, 23%)

Figure 7-5. Effect of increasing span length (box-girder with h=2.5m)

92

7.1.1 Effects of Viewing Points

The visual impact of altering span-to-depth ratio becomes more or less noticeable depending on

the location of the observer. First, as the observer approaches the bridge, the effect of changing

slenderness ratio is more obvious. Figures 7-6 to 7-8 demonstrate the influence of viewing distance

by comparing box-girders with span length of 50m and slenderness ratios of 10 and 20 when

viewed from distances of 300m, 150m, and 75m.

a) L/h=10

b) L/h=20

Figure 7-6. Viewed from 300m

a) L/h=10

b) L/h=20


a) L/h=10

b) L/h=20


93

In addition to viewing distance, the viewing angle also influences the impact of span-to-depth

ratio. Figure 7-9 shows that the pier width-to-height ratio has greater visual impact than the

longitudinal span-to-depth ratio when the bridge is viewed from beneath along the length of the

bridge. This viewing angle is of particular importance if pedestrians can walk below the bridge.

a) L/h=10 with wide wall piers

b) L/h=20 with wide wall piers

c) L/h=10 with narrow piers

d) L/h=20 with narrow piers

Figure 7-9. Effects of pier width-to-height ratio and span-to-depth ratio

94

As the viewing angle becomes less oblique, the impact of pier dimensions diminishes while the

span-to-depth ratio has growing influence on the perceived slenderness of the structure Figure 7-10.

a) L/h=10

b) L/h=20

Figure 7-10. Effect of span-to-depth ratio as viewing angle becomes less oblique

7.1.2 Other Factors that Affect Visual Slenderness

In addition to span-to-depth ratio, other factors also affect the perceived thickness of the

superstructure. These factors include the bridge height, pier configuration, length of deck slab

cantilevers in a box-girder, and railing type. In this section, each factor is investigated using a box-

girder model with a span length of 50m and slenderness ratio of 20.

95

First, Figure 7-11 compares a tall bridge with a low one. Although the span-to-depth ratios are

the same for both bridges, the tall one appears to be more slender due to the larger opening under

the bridge. This large opening contrasts with the slender superstructure, thus reducing the perceived

girder depth. The low bridge, on the other hand, appears heavy because the girder depth is similar to

the bridge height. In fact, to achieve sufficient slenderness and transparency, Menn (1990)

suggested that the bridge height needs to be at least four times greater than the girder depth. This

height suggestion ensures that there is a large contrast between the girder depth and the depth of the

opening under the bridge, thus lowering the perceived thickness of the superstructure.

a) Bridge height/girder depth = 5

b) Bridge height/girder depth = 1.75

Figure 7-11. Effect of bridge height on perceived superstructure slenderness

The pier configuration also affects the visual impact of span-to-depth ratio (Figure 7-12). More

pronounced piers clearly separate the individual span lengths. As a result, the perceived

superstructure thickness still depends on the span-to-depth ratio. Conversely, narrow piers that are

tucked in underneath the superstructure draw less attention and thus accentuate the continuity of the

entire girder. The perceived slenderness is then related to the ratio between the visually

uninterrupted length of the superstructure and the girder depth.

a) Pronounced piers

b) Less obtrusive piers

Figure 7-12. Effect of pier configuration on perceived superstructure slenderness

96

Thirdly, the perceived slenderness of the superstructure is influenced by the length of deck slab

cantilever in box-girders. Figure 7-13 compares a bridge with short cantilevers to a bridge with

longer cantilevers. The latter bridge appears to be more slender because the long cantilevers cast

shadows onto the girder webs and these shadows conceal a portion of the girder depth. The

continuous shadow line along the girder also emphasizes the overall superstructure length. Hence,

slenderness depends more on the bridge length than the span length in this case.

a) Short deck cantilevers

b) Long deck cantilevers

Figure 7-13. Effect of deck cantilever length on perceived superstructure slenderness

Lastly, the railing type contributes to the visual slenderness of the superstructure. Figure 7-14

shows two bridges with open-type metal railings and concrete barrier walls. Any railing increases

the perceived depth of the girder. However, the first case appears more slender because metal

railings are more transparent compared to concrete barriers. Concrete barriers add an extra depth of

solid concrete which represents a major visual bulk. These concrete barriers can be as tall as 1.37m

for highway bridges according to CHBDC (CSA 2006) and this height increases with increasing

traffic volume and speed (Dorton 1991).

a) Open-type metal railings

b) Concrete barrier walls

Figure 7-14. Effect of railing type on perceived superstructure slenderness

97

7.2 Evolution of the Visually Optimal Span-to-Depth Ratio

As demonstrated previously, varying the span-to-depth ratio clearly has an impact on aesthetics.

Yet, visually optimal span-to-depth ratios cannot be easily defined because aesthetics is not a

quantifiable attribute. The ratio that yields the best-looking bridge changes over time and depends

on the background of the observer. This section explores the visually optimal slenderness ratio by

examining the works and design philosophies of various prominent bridge designers.

First, the aesthetically optimal slenderness ratio has changed throughout the history of concrete

bridges. This section traces the development of slenderness in concrete arch and girder bridges.

Arch bridges are considered, because early concrete bridges were mostly arches and the concept of

arch slenderness, which is generally associated with the arch thickness and deck depth with respect

to span length, is analogous to the influence of span-to-depth ratios on the slenderness of girder

bridges. Most importantly, although the representation of slenderness is different for the two bridge

types, both arch and girder bridges demonstrate evident improvement in slenderness due to

economic or aesthetic reasons.

In the 19th century, concrete bridges were generally deep and heavy because concrete was

regarded as artificial masonry and arch was the primary structural form for concrete bridges at that

time. Also, the society preferred the massiveness, rigidity, and embellishments associated with

traditional masonry arches. This trend can be seen in the Glenfinnan Railway Viaduct 1901 which is

one of the first major concrete bridges (Figure 7-14).

Figure 7-14. Glenfinnan Viaduct, 1901 (Cortright 1997)

Improvement in the slenderness of arches was evident by the late 19th century due to the

development of reinforced concrete which allowed arches to be thinner and flatter. Robert Maillart

utilized the new material and created the Stauffacher Bridge 1899 which is a slender three-hinged

arch concealed with stone-cladding and ornaments such that it resembles traditional masonry arches

(Figure 7-15 a). In his subsequent bridges, Maillart abandoned the traditional architectural forms

and began his pursuit for slenderness and simplicity (Figure 7-15). First, he eliminated the stone-

98

cladding in the Zuoz Bridge 1901 to reveal the slender arch. He further enhanced slenderness by

removing the spandrel walls in the Tavanasa Bridge 1906.

a) Stauffacher Bridge, 1899 (Billington 1997)

b) Zuoz Bridge, 1901 (Billington 1990)

c) Tavanasa Bridge, 1906 (Billington 1990)

d) Salginatobel Bridge, 1930 (Billington 1990)

Figure 7-15. Slender bridges by Maillart

One of the major breakthroughs in bridge aesthetics is Maillart’s Salginatobel Bridge 1930

which accentuates the simple three-hinged arch form devoid of any structurally unnecessary

components (Figure 7-15 d). The bridge’s exceptional slenderness contrasts with the traditional

masonry-like structures that were popular at Maillart’s time. The design was chosen mostly for its

economic efficiency instead of its aesthetic value. In fact, its aesthetic merit was not widely

recognized outside of Switzerland until decades after the bridge was constructed. The Salginatobel

Bridge is now praised as a masterpiece of structural art by scholars like Billington (1990) and Bill

(1955).

In the 1930s, the development of reinforced concrete also allowed for the construction of

concrete girder bridges which could reach spans of over 70m. However, long-span reinforced

concrete girders commonly had problems with deformations and cracking. To minimize these

problems, the girders needed to be very deep, with span-to-depth ratios of less than 10 (Menn 1990).

Constant-depth girders with such ratios would be heavy, so long-span reinforced concrete bridges

were usually haunched. Two examples of these variable-depth girders are the Villeneuve-St.

Georges Bridge 1939 and the Waterloo Bridge 1939 which are both haunched box-girders (Figure

7-16). Their span-to-depth ratios at the supports are 9.7 and 10 respectively while the midspan

ratios are 31 and 32 (Menn 1990).

99

Figure 7-16. Waterloo Bridge over the Thames, 1939 (Darger 2002)

Figure 7-17. Changis-sur-Marne Bridge, 1948

(Mossot 2007)

Significant improvement in slenderness for girder bridges only began with the introduction of

prestressing technology to bridge construction in the 1940s. Prestressing enhances structural

behaviour by reducing tensile stresses in the concrete and increasing the load-carrying capacity with

the use of high-strength tendons. As a result, longer and more slender girder bridges were feasible.

Some notable examples of slender prestressed bridges include Eugene Freyssinet’s series of

post-tensioned bridges along the Marne River 1948 (Figure 7-17). These bridges achieved visual

slenderness with the use of prestressing technology, which allowed for longer spans and thinner

decks, and had a midspan span-to-depth ratio of up to 40. This is a large improvement in

slenderness compared to the previous examples of reinforced concrete girders which had a midspan

ratio close to 30. This improvement in slenderness through post-tensioning technology was mainly

driven by material economy, structural efficiency and construction speed.

The aesthetic value associated with slenderness was not fully

appreciated by society until the 1960s which marked the beginning of

an era that praised minimalism and simplicity. The public’s perception

on bridge slenderness was demonstrated in a bridge aesthetics survey

conducted in 1969 by A.G.D. Crouch under the supervision of Colin

O’Connor at the University of Queensland (Crouch 1974, O'Connor

1991). Crouch was an Engineer with the Canberra Department of

Works while O’Connor was a Professor of Civil Engineering and has

published a number of books and papers regarding bridge design. The

bridge aesthetics survey investigated the relative visual merits of

various substructure and superstructure proportion parameters (e.g.

span-to-pier thickness ratio, span-to-column height ratio, etc.) through

the use of simple sketches of bridges with varying proportions. The

survey sample included 170 civil engineers, architects, and people with

Figure 7-18. Sketches to

evaluate aesthetic impact of

span-to-depth ratios

(O'Connor 1991)

100

no education in structural design (a control group). The civil and architecture groups consisted of

students, university staffs as well as people in practice. One particular superstructure proportion

parameter that was investigated was the span-to-depth ratio (Figure 7-18). The response

corresponded to a preferred ratio of 34.3 for engineers, 20.5 for architects and 24.4 for the control

group. The difference among the three groups demonstrated that the visually optimal ratio depends

on the background of the observer. Engineers preferred a higher ratio because they valued material

efficiency. The other groups, in contrast, preferred a deeper girder because they saw depth as a sign

of strength and visual elegance as a matter of good proportions. The sketches used in this survey

might be too simple to confidently yield specific values for visually-optimal span-to-depth ratios.

Nonetheless, the survey indicated that people in general preferred a more slender structure with a

minimum span-to-depth ratio of 20 (i.e. the top two sketches in Figure 7-18).

Since there was greater public resonance for visual slenderness, higher span-to-depth ratios

gained unprecedented popularity in bridge designs in the 60s and 70s. Bridge designers, therefore,

consciously pursued slenderness based on its aesthetic merits in addition to its material economy. In

particular, two renowned bridge engineers designed a number of girder bridges with exceptional

slenderness at this time: Leonhardt and Menn. Leonhardt believed that “the slender bridge looks

better than the clumsy one. A slender look is therefore a design feature well worth striving for”

(Leonhardt 1982). He also claimed that heavy bridges appear “depressing” whereas lighter bridges

are more elegant. His design philosophy was demonstrated in the Neckar Viaduct 1977 and the

Kocher Valley Viaduct 1979 (Figures 7-19 and 7-20). The former haunched girder has a

slenderness ratio of 56 at midspan and 25 at the supports while the latter constant-depth girder has a

ratio of 21.2. Both the haunched and constant-depth girders demonstrate significant improvement in

slenderness compared to bridges from the 30s which used span-to-depth ratios of 30 at midspan and

10 at the supports for haunched girders and a ratio of 10 for constant-depth girders.

Figure 7-19. Neckar Valley Viaduct, 1977 (Leonhardt

1982)

Figure 7-20. Kocher Valley Viaduct, 1979 (Leonhardt

1982)

101

Menn, on the other hand, believed that visual elegance was related to the efficient use of

material which could be demonstrated by slenderness and lightness of a structure (Menn 1990).

Two of his designs that incorporated this concept of visual elegance are the Pregorda Bridge 1974

and the Felsenau Bridge 1974 (Figures 7-21 and 7-22). Like Leonhardt’s bridges, these two bridges

are considerably more slender than the girder bridges constructed in the previous decades. The

Pregorda Bridge is a constant-depth girder with a slenderness ratio of 22.2 while the main spans of

the Felsenau Bridge are haunched and has ratios of 48 at midspan and 18 at the supports.

Figure 7-21. Pregorda Bridge, 1974 (Menn)

Figure 7-22. Felsenau Bridge, 1974 (Menn)

Other contemporary bridge experts are also strong advocates of bridge slenderness and they

have expressed the importance of slenderness in bridge aesthetics in a number of publications. One

of these bridge professionals is Edward Wasserman who is the Civil Engineering Director of the

Structures Division for the Tennessee Department of Transportation and has been involved in the

design of over 2200 bridges. He claimed that a span-to-depth ratio between 25 and 30 would

produce a well-proportioned superstructure that “appears to float gracefully” whereas a lower ratio

would result in a bridge that appears to “loom heavily upon the landscape” (Wasserman 1991).

Likewise, Arthur Elliott, Bridge Engineer responsible for all bridge planning and design with the

California Department of Transportation from 1953 to 1973, stated that a “blocky, heavy, and

poorly proportioned” bridge is simply not beautiful (Elliott 1991). Frederick Gottemoeller, who has

produced a number of publications regarding bridge aesthetics and has developed the aesthetic

design guidelines for Maryland and Ohio, also recognizes the visual benefits of slenderness. He

claimed that better-looking bridges are characterized by their “simplicity, thinness, and continuity”

(Gottemoeller 2004).

As shown in this historical study, the public perception of bridge aesthetics changes over time.

In the early 1900s, people preferred heavier bridges whereas slender bridges were appreciated in the

60s and 70s. Although the visually optimal span-to-depth ratio cannot be easily determined,

contemporary bridge designers generally favour a higher ratio and regard slenderness as a key

element in a good-looking bridge.

102


This chapter demonstrates that span-to-depth ratio has direct impact on perceived superstructure

slenderness by comparing 3-D drawings of bridges with varying ratios. Bridges with the most cost-

effective ratio are visually more slender than those with conventional ratios. The visual difference

between the cost-effective ratio and conventional ratio is particularly noticeable for the cast-in-place

on falsework and precast span-by-span box-girders. For cast-in-place solid slab, the difference is

less obvious because the conventional ratio is already high (i.e. 30) and increasing the ratio beyond

25 is found to have negligible visual impact. This chapter further shows that the effect of varying

span-to-depth ratio reduces as the observer moves away from the bridge or as the viewing angle

becomes more oblique. Factors, such as low and protruding piers, short deck cantilever lengths, and

thick concrete railing, also reduce the perceived slenderness. Lastly, a historical study indicates that

the visually optimal slenderness ratio evolves over time and contemporary bridge engineers

appreciate the aesthetic merit of superstructure slenderness.

103

8 CONCLUSIONS

Girder-type bridges have commonly been designed using conventional span-to-depth ratios

which have not changed significantly despite recent development in material strengths and

construction technologies. This study determines the optimum slenderness ratios for three types of

girder bridges constructed with high-strength concrete: cast-in-place on falsework box-girder and

solid slab, and precast segmental span-by-span box-girder. The ratios are optimized based on

material consumption and total construction cost criteria. Aesthetic comparisons are also performed

to determine the visual impact of these optimum ratios. The primary results of this thesis are

summarized as follows.

8.1 Conventional Span-to-Depth Ratios

A study of 86 constant-depth girder bridges reveals that the typical ranges of span-to-depth

ratios are 17.7 to 22.6 for cast-in-place box-girder, 19 to 35 for cast-in-place voided slab, 22 to 39

for cast-in-place solid slab, and 15.7 to 18.8 for precast segmental box-girder. The study

demonstrates that the ratios for cast-in-place and precast segmental box-girders have not varied

significantly from 1958 to 2007. The study also indicates that cast-in-place slabs constructed after

1975 are mostly voided slabs with slenderness ratios below 25 due to the more stringent code

requirements in recent years.

8.2 Maximum Span-to-Depth Ratios

The maximum span-to-depth ratio, which satisfies safety, serviceability, and constructability

requirements, varies with bridge type and span length. For cast-in-place box-girder with spans of

35m, 50m, and 60m, the maximum ratios are 25, 30, and 35 respectively. These values are restricted

by the interior box cavity height requirement which is necessary to provide sufficient space for

workers. The maximum ratio is also 35 for the case with a span of 75m; it is limited by the number

of tendons that can fit inside the webs of the box section.

The maximum ratios for solid slab are 40 for spans of 20m and 25m, and 45 for spans of 30m

and 35m. These ratios are governed by the maximum reinforcement criterion which ensures

adequate ductile behaviour. This limitation is especially critical for solid slabs, because slabs have

heavier dead loads compared to box-girders with the same ratio, so more reinforcements are needed.

For precast segmental box-girder, the maximum ratios are 20 for a span of 30m, and 25 for

spans of 40m and 50m. Like cast-in-place box-girders, these ratios are limited by the minimum

height requirement for the interior box cavity. Even though the governing factor of maximum

slenderness ratio is the same for both bridge types, the ratios for precast box-girders are lower,

because external tendons are used, further reducing the height of the cavity within box-girders.

104

8.3 Material Consumption Comparisons

As slenderness increases, the prestressing and concrete strength demands increase while

concrete volume decreases. For both cast-in-place and precast segmental box-girders, the decrease

in volume is small compared to the increase in prestressing since only a small amount of concrete at

the webs is eliminated while the moment resistance is significantly lowered as the ratio increases.

Reduction in concrete volume further diminishes for higher ratios because the bottom slab thickness

needs to be increased to accommodate the larger compressive force in a slender girder. This

increase in slab thickness counteracts the decrease in web volume, thus the reduction in concrete

volume becomes less for higher ratios. Moreover, a concrete compressive strength of 50 MPa is

sufficient to satisfy design requirements in every analysis case for both of these bridge types.

Decrease in concrete volume in a solid slab is more significant since a large strip of concrete as

wide as the soffit is removed as the ratio increases. Therefore, this volume reduction is proportional

to the decrease in girder depth. On the other hand, prestressing demand increases with slenderness.

This increase in prestressing is less than the one for cast-in-place box-girders, because solid slabs

can accommodate many tendons at the same elevation while tendons in box-girders need to be

placed in multiple layers within the webs which reduces the prestressing efficiency. Also, higher

strength concretes (i.e. f’c = 50 to 80 MPa) are used for slender slabs to satisfy ductility requirement

by lowering the prestressing demand. As a result, prestressing consumption in solid slabs increases

with span-to-depth ratio at a slower rate relative to box-girders.

Material consumptions are compared on the basis of material costs. Considering only the pure

material costs without the cost of formworks, falsework or precasting operations, the most efficient

ratios are15 for both cast-in-place and precast box-girders and 30 for solid slabs. If the costs of

concrete fabrication and placement are included, the most cost-efficient ratios for the three bridge

types increase to 25, 40, and 20. This increase in cost-optimal ratios indicates that the construction

economy related to a slenderer and lighter structure is a crucial aspect in the optimization of span-

to-depth ratio. More importantly, over the entire range of ratios investigated (i.e. 10 to 35 for cast-

in-place box-girder, 30 to 45 for cast-in-place solid slab, and 15 to 25 for precast segmental box-

girder), the maximum variations in pure material cost are 42%, 28%, and 12% while the maximum

variations in total superstructure cost are only 20%, 12%, and 3.3% for each bridge type.

8.4 Total Construction Cost Comparisons

The total construction costs are computed assuming the superstructure accounts for 54.5% of

total construction cost. With the additional costs of mobilization, substructure, and accessories, the

most cost-efficient ratios remain the same as the ones based on superstructure costs only. However,

105

the economy of using these optimal ratios diminishes when the total construction costs are

considered; the maximum savings within the analysis range of ratios are reduced to less than 11%,

6.2%, and 1.8% for cast-in-place box-girder, cast-in-place solid slab, and precast segmental box-

girder respectively. This finding indicates that optimizing one particular structural component (i.e.

superstructure span-to-depth ratio) does not result in significant economy. Greater cost savings

emerge from the selection of an appropriate bridge type. For instance, within the same range of span

lengths and span-to-depth ratios, a cast-in-place on falsework box-girder is more economical than a

solid slab. Moreover, these findings are not sensitive to changes in material unit prices and in total

construction cost breakdown; the cost-optimal ratios remain the same regardless of these changes.

The variability in total construction cost over the analysis range of ratios is less than 13% when the

material unit price is altered by 50% or when the superstructure cost contribution in the total cost

breakdown rises from 54.5% to 80%.

8.5 Aesthetic Comparisons

Since the total construction cost does not vary significantly over the entire range of span-to-

depth ratios investigated, the designer has more freedom to select the slenderness ratio without

much economic constraints. Varying the ratio changes the superstructure slenderness which is

generally the most important visual component of a girder-type highway overpass. A historical

study indicates that although the public perception on slenderness has evolved over time,

contemporary bridge engineers, especially the ones from the 1960s and 1970s, recognize and

appreciate the aesthetic merit of slenderness associated with high span-to-depth ratios.

The visual impact of span-to-depth ratios is examined by comparing 3-D renderings of bridges

with different ratios. This comparison determines that using cost-optimal ratios instead of

conventional ones would result in considerable enhancement in the superstructure slenderness for

cast-in-place and precast box-girders. Yet, the aesthetic impact of using the optimal ratio in solid

slabs is negligible because increasing the ratio beyond 25 is found to have no apparent visual

difference. Furthermore, the visual effects of varying span-to-depth ratio reduce as the observer

moves away from the bridge or as the viewing angle becomes more oblique. In addition to lowering

the slenderness ratio, other factors that reduce perceived slenderness include a low bridge height,

protruding piers, short deck cantilever lengths, as well as solid concrete railing.

8.6 Optimal Span-to-Depth Ratios

Based on construction economy and aesthetics considerations, the optimal span-to-depth ratios

are established to be 25, 40, and 20 for cast-in-place on falsework box-girder, cast-in-place on

falsework solid slab, and precast segmental span-by-span box-girder. However, these optimums are

106

expected to change in a real situation because they are determined based on specific parameters

defined for this study such as cross-section dimensions and span arrangements.

More importantly, the study demonstrates that, within the analysis range of ratios, the total

construction cost is relatively insensitive to changes in the ratio. This finding is valid even if the

optimum ratio changes, because the study shows that cost savings from optimizing span-to-depth

ratio are generally minor compared to other construction cost components. The results of this study

indicate that, compared to conventional ratios defined for normal-strength concrete bridges, a

greater range of values can be used without significant cost premiums for high-strength concrete

bridges (i.e. f’c = 50 to 80 MPa): 10 to 35 for cast-in-place on falsework box-girder; 30 to 45 for

cast-in-place on falsework solid slab; 15 to 25 for precast segmental span-by-span box-girder.

107

REFERENCE

AASHTO-PCI-ASBI. (1997)."Segmental Box Girder Standards for Span-by-Span and Balanced

Cantilever Construction." American Segmental Bridge Institute, <www.asbi-assoc.org>

(Feb. 1, 2009).

AASHTO. (1994). AASHTO LRFD Bridge Design Specifications, American Association of State

Highway and Transportation Officials, Washington, D.C.

ACI-ASCE Committee 343. (1988). "Analysis and design of reinforced concrete bridge structures."

American Concrete Institute, Detroit.

Belli, A. (2003). "Interstate 895 Bridge, James River, Richmond, Virginia, USA." Structural

Engineering International:Journal of the International Association for Bridge and

Structural Engineering (IABSE), 13(3), 177-179.

Bennett, M. V., and Taylor, A. J. (2002). "Woronora River Bridge, Sydney." Structural

Engineering International: Journal of the International Association for Bridge and

Structural Engineering (IABSE), 12(1), 28-31.

Bill, M. (1955). Robert Maillart, Girsberger Zürich.

Billington, D. (1990). Robert Maillart and the Art of Reinforced Concrete, The MIT Press,

Cambridge.

Billington, D. (1997). Robert Maillart: Builder, Designer, and Artist, Cambridge University Press,

Cambridge.

Cao, S., Wu, W., Tu, J., and Wu, E. (2006). "Design & Construction of Ngong Shuen Chau

Viaduct." International Conference on Bridge Engineering-Challenges in the 21st Century,

Hong Kong.

Chen, W.-F., and Duan, L. (1999). Bridge Engineering Handbook, CRC Press, Boca Raton.

Collins, M. P., and Mitchell, D. (1997). Prestressed Concrete Structures, Response Publications,

Toronto.

Cortright, R. (1997)."Glenfinnan Viaduct." Nicolas Janberg's Structurae,

<en.structurae.de/photos/index.cfm?JS=54616> (June 1, 2009).

Cremer, J.-M., Counasse, C., and Delforno, J.-Y. (2003). "The Sart Canal-Bridge, houdeng -

Aimeries, Belgium." Structural Engineering International: Journal of the International

Association for Bridge and Structural Engineering (IABSE), 13(1), 19-22.

Crouch, A. G. D. (1974). "Bridge Aesthetics - A Sociological Approach." Civil Engineering

Transactions, CE16(2), 138 - 142.

CSA (1982). "Steel for Prestressed Concrete Tendons." CSA G279-M1982, Canadian Standards

Association, Rexdale.

CSA. (2006). Canadian Highway Bridge Design Code, C. S. Association, ed., Canadian Standards

Association, Mississauga, Ontario.

http://www.asbi-assoc.org/

108

Darger, N. J. (2002)."Waterloo Bridge (1945)." Nicolas Janberg's Structurae,

<en.structurae.de/photos/index.cfm?JS=3502> (June 1, 2009).

Dorton, R. A. (1991). "Aesthetic Considerations for Bridge Overpass Design." Bridge aesthetics

around the world, Transportation Research Board, National Research Council, Washington,

D.C., 10-17.

DSI. (2008). "DYWIDAG Post-Tensioning Systems." D.-S. International, ed., DYWIDAG-Systems

International.

Duan, L., Chen, K., and Tan, A. (1999). "Design Practice in Europe." Bridge Engineering

Handbook, W.-F. Chen and L. Duan, eds., CRC Press Boca Raton, 10-11 to 10-37.

Dufferin Concrete. (2009)."2009 Price List." Dufferin Concrete, <www.dufferinconcrete.ca> (June

1, 2009).

Elliott, A. L. (1991). "Creating a Beautiful Bridge." Bridge aesthetics around the world

Transportation Research Board, National Research Council, Washington, D.C., 215-229.

Gauvreau, P. (2006). "Bridges." Post-Tensioning Manual, Post-Tensioning Institute, Phoenix, 195-

240.

Gottemoeller, F. (2004). Bridgescape : The Art of Designing Bridges, Wiley, Hoboken.

Hassanain, M. (2002). "Optimal Design of High-Performance Concrete Adjacent Precast Box

Girder Bridges." 6th International Conference on Short & Medium Span

Bridges:Developments in Short & Medium Span Bridge Engineering, CSCE, Vancouver.

Hewson, N. R. (2003). Prestressed Concrete Bridges : Design and Construction, Thomas Telford,

London.

Hillerborg, A. (1996). Strip Method Design Handbook, E & FN Spon, London.

Holgate, A. (1996). The Art of Structural Engineering: The Work of Jorg Schlaich and His Team,

Ed. Axel Menges, Stuttgart.

Holowka, M. (1979). "Deck Slab Investigation of a Voided Posttensioned Concrete Bridge Deck -

Bridge 20." Ontario Ministry of Transportation and Communications, St. Catharines.

Janberg, N. (2009)."Structurae." <en.structurae.de> (June 1, 2009).

Kosmatka, S. H., Kerkhoff, B., Panarese, W. C., MacLeod, N. F., and McGrath, R. J. (2002).

Design and Control of Concrete Mixtures, Cement Association of Canada, Ottawa.

Leonhardt, F. (1979). Grundlagen des Massivbrückenbaues, Springer-Verlag, Berlin/Heidelberg.

Leonhardt, F. (1982). Brücken : Ästhetik und Gestaltung = Bridges: Aesthetics and Design,

Deutsche Verlags-Anstalt, Stuttgart.

Linse, A., and Wössner, K. (1978). "Kochertalbrücke-Entwürfe einer Grossbrücke. " Bauingenieur,

53(12), 453-463.

http://www.dufferinconcrete.ca/

109

Liu, X. P., Qin, Z. P., Leung, Y. W., and Yue, L. (2007). "Construction of the precast segmental

approach structures for Sutong Bridge." 5th International Conference on Current and

Future Trends in Bridge Design, Construction and Maintenance, Thomas Telford Ltd,

Beijing.

MacGregor, J., and Bartlett, F. M. (2000). Reinforced Concrete: Mechanics and Design, Prentice

Hall Canada, Scarborough.

Meades, P., and Green, R. (1974). "Model Studies for Voided Posttensioned Concrete Slab

Bridges." Ministry of Transportation and Communications, Ontario, Toronto.

Menn, C. Christian Menn, Consulting Engineer: Projects." <www.christian-menn.ch> (July 1,

2009).

Menn, C. (1990). Prestressed Concrete Bridges P. Gauvreau, ed., P. Gauvreau, translator,

Birkhäuser Verlag, Basel.

Mossot, J. (2007)."Changis-sur-Marne Bridge " Nicolas Janberg's Structurae, (June 1, 2009).

MTO (2003). "Structural Manual." Ontario Ministry of Transportation, Bridge Office, St.

Catharines.

Muller, J. (1999). "Design Practice in Europe." Bridge Engineering Handbook, W.-F. Chen and L.

Duan, eds., CRC Press Boca Raton, 64-61 to 64-37.

NRS. (2008)."Gantries-Guangzhou LRT Project." NRS Bridge Construction Equipment,

<http://www.nrsas.com/v2009/project/pro_gantries.php> (June 1, 2009).

O'Brien, E. J., and Keogh, D. L. (1999). Bridge Deck Analysis, E & FN Spon, London.

O'Connor, C. (1991). "Empirical Assessment of Bridge Aesthetics: An Australian View." Bridge

Aesthetics Around the World, Transportation Research Board, National Research Council,

Washington, D.C., 230-240.

OHBDC. (1983). Ontario Highway Bridge Design Code, Highway Engineering Division, Toronto.

OPSS. (2007). Material Specification for Concrete - Materials and Production, Ontario Provincial

Standard Specification.

OSHA. (2006)."Overhead Launching Gantry Crane Safety and Health Information Bulletin."

Occupational Safety & Health Administration, <www.osha.gov/dts/shib/shib050106.html>

(June 1, 2009).

Park, R., and Gamble, W. L. (2000). Reinforced Concrete Slabs, Wiley, New York.

PEER. (2005)."Neckar Valley Viaduct, Weitingen, Germany (1978)." Pacific Earthquake

Engineering Research Center, University of California,

<http://nisee.berkeley.edu/elibrary/getpkg?id=GoddenG43-50> (July 1, 2009).

Sauvageot, G. (1999). "Segmental Concrete Bridges." Bridge Engineering Handbook, W.-F. Chen

and L. Duan, eds., CRC Press, Boca Raton.

http://www.christian-menn.ch/

http://www.nrsas.com/v2009/project/pro_gantries.php

http://www.osha.gov/dts/shib/shib050106.html

http://nisee.berkeley.edu/elibrary/getpkg?id=GoddenG43-50

110

Scollard, C. R., and Bartlett, F. M. (2004). "Rehabilitation Criteria for Post-Tensioned Voided-Slab

Bridges." Canadian Journal of Civil Engineering, 31(6), 977-987.

SNC-Lavalin. (2008). "Highway 400 Major MacKenzie Drive to Highway 11 Heritage Bridge

Replacement Design Concept Study." Report prepared by SNC-Lavalin, submitted to

Ontario Ministry of Transportation.

TRB. (1990). Truck Weight Limits: Issues and Options, Transportation Research Board Washington,

D.C.

Vogel, T. (1997). Christian Menn-Brückenbauer, Birkhäuser Verlag, Basel, 57.

Wasserman, E. P. (1991). "Aesthetics for Short- and Medium-Span Bridges." Bridge aesthetics

around the world Transportation Research Board, National Research Council, Washington,

D.C., 58-66.

Weinlandbrücke (1958). "Die Weinlandbrücke in der Umfahrungsstrasse von Andelfingen."

Direktion der öffentlichen Bauten des Kantons Zürich (Administration of Public Buildings

of the Canton of Zürich).

111

APPENDIX A: CHAPTER 2 SUPPLEMENTARY INFORMATION

11

2

A.1 Cast-in-Place on Falsework Box-Girder

Table A-1. Cross-sectional drawings for cast-in-place on falsework box-girders in Chapter 2

Bridge no. Cross-section Bridge no. Cross-section Bridge no. Cross-section

1

13

23 & 24

3

14

25

4

15

26

5 & 6

16

27

7

17

28

8

18

29

9

19

31

10

20

32

11

21

33

12

22

34

11

3

Table A-1. Cross-sectional drawings for cast-in-place on falsework box-girders in Chapter 2 (continued)


35

38

42

36

40

43

37

41

44

Table A-2. Cast-in-place on falsework box-girders in Chapter 2

Bridge no.

Name Location Completion year

Construction method

Span arrangement (m) Span length L (m)

Depth h (m)

Span-to-depth ratio L/h

Reference

1 Grenz Bridge at Basel

Switzerland 35.4 2 17.7 Drawing from Gauvreau

2 Sart Canal-Bridge Belgium 2002 Incremental launching

13 spans 36 3 12 (Cremer et al. 2003)

3 Weyermannshaus Bridge

Switzerland 37.75 2 18.88 Drawing from Gauvreau

4 Eastbound Walnut Viaduct

U.S.A. 1986 CIP on falsework

24.4+35.4+38.1+ 38.1+34.4+37.16+ 38.1+38.6+38.6+ 34.0+24.7 = 382

38.6 1.68 23.0 Designer: T.Y. Lin International for Department of Highways, Colorado Drawing from Gauvreau

5 Taiwan High Speed Rail (1)

Taiwan 2000 CIP span-by-span

3*25+25.5+40+25+2*30+3*25 = 300.5

40 3.5 11.4 Drawing from Gauvreau

6 Taiwan High Speed Rail (2)

Taiwan 2000 CIP span-by-span

25+35+26.5+25+ 40+24.5+26.5+4* (30) = 322.5


7 Pregorda Bridge Switzerland 1974 Span-by-span on falsework

40 1.8 22.2 Designer: Christian Menn Drawing from Gauvreau (Vogel 1997)

8 Almese Viaduct Italy 1990 Balanced cantilever

2*(21+ 3*40 + 20.7) = 323.4

40 2.2 18.2 DEAL Job Report (Segmental Bridge Data)

9 Condove Viaduct Italy 1992 Balanced cantilever

2*(20.65+17*40+21.15) = 1443.6


11

4

Table A-2. Cast-in-place on falsework box-girders in Chapter 2 (continued)

Bridge no.


Construction method


Depth h (m)


Reference

10 Gravio Viaduct Italy 1992 Balanced cantilever

2*(20.65+12*40+21.15) = 1043.6


11 Borgone Viaduct Italy 1992 Balanced cantilever

2*(19.40+26*40+19.15) = 2157.1


12 Quadinei Bridge Switzerland 1967 Span-by-span on falsework

40 2 20 (Menn 1990)

13 Altstetter Viaduct Switzerland 41 1.9 21.6 Drawing from Gauvreau

14 Reuss Bridge Switzerland 1972 CIP on falsework

28.1+34+37.5+40*3+42.5+40*2+37.5+32.5 = 412

42.5 2.4 17.7 Drawing from Gauvreau Designer: Christian Menn

15 Cerchiara Viaduct Italy 1992 Balanced cantilever

40.30+28*42.50+34= 1264.3

42.5 2.3 18.5 DEAL Job Report (Segmental Bridge Data)

16 Castello Viaduct Italy 1992 Balanced cantilever

40.30+6*42.50+ 36 = 331.3


17 Costacole Viaduct Italy 1992 Balanced cantilever

34+5*42.50+34 = 280.50 42.5 2.3 18.5 DEAL Job Report (Segmental Bridge Data)

18 Ferroviario Overpass at Bolzano

Italy 1974 Spans = 36.7 to 45 Total length = 134


19 Krebsbachtal Bridge

Germany 1975 Incremental launching

45 3.49 12.9 (Menn 1990)

20 Shatt Al Arab Bridge

Iraq 1978 Incremental launching

Spans = 38.25 to 46.90 Total length = 761

46.9 3.65 12.8 Drawing from Gauvreau

21 Ancona Viaduct Italy Segmental Spans = 21.76 to 47.50 Total length = 2015.9


22 Felsenau Bridge (approaches)

Switzerland 1975 Span-by-span on falsework

48*5 = 240 48.0 3.0 16.0 Designer: Christian Menn (Menn 1990)

23 La Molletta Viaduct Italy 1988 Segmental (40+2*50+40) +(40+2*50+40) = 360


24 Fosso Capaldo Viaduct

Italy 1988 Segmental (40+11*50+40) +(40+11*50+40) = 1260


25 Sihlhochstrasse Bridge

Switzerland 54.5 1.85 29.5 Drawing from Gauvreau

26 Grosotto Viaduct Italy 1992 Balanced cantilever

45+15*55+45= 915 55 2.75 20 DEAL Job Report (Segmental Bridge Data)

27 Grosio Viaduct Italy 1994 Balanced cantilever

45+24*55+45 = 1410 55 2.75 20 DEAL Job Report (Segmental Bridge Data)

28 Tiolo Viaduct Italy Balanced cantilever

45+9*55+45 = 585 55 2.75 20 DEAL Job Report (Segmental Bridge Data)

29 Denny Creek U.S.A. 1980 Spans = 57.32 57.32 2.744 20.9 Drawing from Gauvreau

11

5

Table A-2. Cast-in-place on falsework box-girders in Chapter 2 (continued)

Bridge no.


Construction method


Depth h (m)


Reference

30 Woronora River Bridge

Australia 2001 Incremental launching

Total length = 521 58.7 4 14.7 (Bennett and Taylor 2002)

31 Valentino Viaduct Italy 1990 Balanced cantilever

30.80+6*60+31 = 421.8 60 3 20 DEAL Job Report (Segmental Bridge Data)

32 Giaglione Viaduct Italy 1992 Balanced cantilever

2*(30.9+9*60+31.05) = 1143.90

60 3 20 DEAL Job Report (Segmental Bridge Data)

33 Venaus Viaduct Italy 1992 Balanced cantilever

(30.7+9*60+ 30.9) + (30.95 +7*60+ 30.7) = 1083.25


34 Passeggeri Viaduct Italy 1992 Segmental (30.9+5*60+31.05)+(30.9+4*60+31.05) = 663.90


35 Brunetta Viaduct Italy 1992 Segmental (30.85+11*60+31.30)+(31.05+ 9*60+59.05) = 1352.25


36 Pietrastretta Viaduct

Italy 1992 Segmental (30.70+4*60+ 30.80)+(30.70+3*60+30.90) = 543.10


37 Deveys Viaduct Italy 1992 Segmental (30.70+3*60+ 30.70)+(30.70+4*60+31.70) = 542.80


38 Gruyère Lake Viaduct

Switzerland 1978 CIP span-by-span

Total length = 2043 60.48 4 15.12 Drawing from Gauvreau

39 Interstate 895 Bridge over James River (approaches)

U.S.A. 2002 Balanced cantilever

64 3 21.3 (Belli 2003)

40 Lätten Bridge Switzerland 65 3.6 18.1 Drawing from Gauvreau

41 Savona Mollere Viaduct

Italy 1994 Segmental 35.3+67.5+36.5+67.5+35.3 = 242.1

67.5 3 22.5 DEAL Job Report (Segmental Bridge Data)

42 Ruina Viaduct Italy 1984 Spans = 44.5 to 73.5 Total length = 785


43 Weinland Bridge Switzerland 1958 Span-by-span on falsework

57+76+88+66 = 287 88 3.9 22.6 (Menn 1990, Weinlandbrücke 1958)

44 Kocher Valley Bridge

Germany 1971 Balanced cantilever

81+7*138+81=1128 138 6.5 21.2 Designer: Fritz Leonhardt (Linse and Wössner 1978, PEER 2005) Drawing from Gauvreau

11

6

A.2 Cast-in-Place on Falsework Solid Slab

Table A-3. Cross-sectional drawings for cast-in-place on falsework solid slabs in Chapter 2

Bridge no. Cross-section Bridge no. Cross-section Bridge no. Cross-section Bridge no. Cross-section

46

51

64

70

47

58

65

71

48

60

66

72

49

61

67

50

62

68

Table A-4. Cast-in-place on falsework solid slabs in Chapter 2

Bridge no.


Construction method


Depth h (m)


Reference

45 Khandeshwar Bridge India 2000 N/A 13.2 0.65 20.3 Designer: Shirish Patel & Associates Consultants Private Limited (Janberg 2009)


Canada 1967 CIP on falsework 12.192+15.24+12.192 = 39.624

15.24 0.686 22.2 Department of Highways Ontario

47 & 48

Spadina Ave. Bridge #18A & B, Hwy 401






50 & 51

Spadina Ave. Bridge #21A & B, Hwy 401



52 Sindelfingen Footbridge Germany 1986 N/A 16.7 0.3 55.7 Designers: Schlaich, Bergermann und Partner (Holgate 1996, Janberg 2009)

53 L 333 Overpass at Bassum Germany 1993 N/A 13.5+16.87+13.5 16.87 0.77 21.9 Designer: Haas Consult and Ingenieurbüro Perlebery (Janberg 2009)

11

7

Table A-4. Cast-in-place on falsework solid slabs in Chapter 2 (continued)

Bridge no.


Construction method


Depth h (m)


Reference

54 Waiblingen Footbridge Germany 1996 N/A 17 0.4 42.5 Designer: Fischer + Friedrich Beratende Ingenieure (Janberg 2009)

55 Mako Bridge Senegal 1975 Incremental launching

16+3*21+16+3*21+16 21 0.88 23.9 Designer: SFEDTP (Janberg 2009)

56 Kittelbaches Bridge Germany 1992 N/A 21.65 1.12 19.3 Designer: Metz-Herder-Wendt und Billig, Schüßler-Plan Ingenieurgesellschaft mbH (Janberg 2009)

57 San Francisco Airport Viaduct

U.S.A. N/A Spans = 18.3 to 22.25, Total length =1630


58 St. Vincent Street Overpass

Canada 1964 CIP on falsework 19.5+2*24.4+19.5 = 87.8 24.4 0.686 35.6 Department of Highways Ontario Drawing from Gauvreau

59 Bridge across Jan-Wellen-Platz

Germany 1961 N/A Total length = 536 25 1 25 Designer: Fritz Leonhardt (Leonhardt 1982, PEER 2005)

60 & 61

Spadina Ave. Bridge #14 & #15, Hwy 401

Canada 1963 CIP on falsework 18.2+25.7+2*23.4+29. +18.2 = 138

29.4 0.762 38.6 Department of Highways Ontario Drawing from Gauvreau


Canada 1963 CIP on falsework 18.2+26.0+24.0+23.4+29.4+18.2 = 139


63 Saale Bridge at Rudolphstein

Germany 1994 N/A 23.4+8*31.2+23.4 31.2 1.56 20 (Janberg 2009)

64 Bridge #20 at Hwy 401/427 Interchange

Canada CIP on falsework 16.4+23.2+21.9+31.4+24.4= 117

31.4 0.99 31.7 (Holowka 1979)


Canada 1963 CIP on falsework 15.24+2*32+23.17+32+15.2 = 149.655

32 0.914 35 Department of Highways Ontario Drawing from Gauvreau

66 Spadina Ave. Bridge #11 Hwy 401

Canada 1963 CIP on falsework

19.96+34.08+29.56+27.88+36.57+34.13+25.50+26.43+38.1+2*35.05+36.27+36.2+36.5+25.603 = 477

38.1 1.27 30 Department of Highways Ontario Drawing from Gauvreau


Canada 1963 CIP on falsework 18.2+38.7+23.1+20.2+34.4+18.2 = 153



Canada 1963 CIP on falsework 27.4+36.5+39.6+27.432 = 131


69 Hundschipfen Bridge Switzerland 2000 N/A 40.465 40.46 1 40.5 Designer: Wolfgang Linder (Janberg 2009)

70 McCowan Road Underpass Canada CIP on falsework 36+45.1+36=117 45.1 1.52 29.7 (Meades and Green 1974)







11

8

A.3 Precast Segmental Span-by-Span Box-Girder

Table A-5. Cross-sectional drawings for precast segmental span-by-span box-girders in Chapter 2


73

78

83

75

79

84

76

80 & 81

85

77

82

86

Table A-6. Precast segmental span-by-span box-girders in Chapter 2

Bridge no.


Construction method


Depth h (m)


Reference

73 Bukit Panjang LRT System 801

Singapore 1998

Segmental span-by-span Total length =7715.24 34 2.16 15.7 DEAL Job Report (Segmental Bridge Data)

74 Wiscasset Bridge U.S.A. 1981

Segmental with launching girder

27.5+21*36.85+27.5 = 829 36.85 2.1 17.5 (Janberg 2009)

75 Chiovano Viaduct Italy 1993 Segmental 39+18*39+39 = 780 39 2.3 17 DEAL Job Report (Segmental Bridge Data)

76 Collecastino Viaduct Italy 1991 Segmental 39+25*39+39 = 1053 39 2.3 17 DEAL Job Report (Segmental Bridge Data)

77 Fiumetto Viaduct Italy 1992 Segmental 39+11*39+39 = 507 39 2.3 17 DEAL Job Report (Segmental Bridge Data)

78 San Leonardo Viaduct Italy 1992 Segmental 39+5*39+39 = 273 39 2.3 17 DEAL Job Report (Segmental Bridge Data)

79 Petto Viaduct Italy 1992 Segmental 39+11*39+39 = 507 39 2.3 17 DEAL Job Report (Segmental Bridge Data)

80 Cadramazzo Viaduct Italy 1985 Balanced cantilever 2*(35+13*40+35) = 1180 40 2.4 16.67 DEAL Job Report (Segmental Bridge Data)

81 Fella IX Viaduct Italy 1986

Balanced cantilever 2*(41+18*40+40.80) = 1604 40 2.4 16.7 DEAL Job Report (Segmental Bridge Data)

82 Malborghetto Viaduct Italy 1986

Balanced cantilever 2*(40+16*45+40) = 1604 45 2.8 16.1

DEAL Job Report (Segmental Bridge Data)

83 Val Freghizia Viaduct Italy 1988 Balanced cantilever (36+15*47+36) *2 = 1554 47 2.8 16.8 DEAL Job Report (Segmental Bridge Data)

84 Fella IV Viaduct Italy 1985 Balanced cantilever 2*(45+6*60+45) = 960 60 3.4 17.6 DEAL Job Report (Segmental Bridge Data)

85 Ngong Shuen Chau Viaduct China 2007 Balanced cantilever 45+4*60+45 60 3.2 18.75 (Cao et al. 2006)

86 Sutong Bridge Approach (Nantong side)

China 2007

Balanced cantilever 50+19*75 75 4 18.75 (Liu et al. 2007)

119

APPENDIX B: SUPPORTING CALCULATIONS

B.1 Flexural Strength for Bonded Tendons at ULS

This calculation is performed for a cast-in-place on falsework box-girder analysis case with span length L of 50m and span-to-depth ratio L/h of 20.

MULS = moment demand = 53 000 kNm

Concrete properties f’c = 50 MPa fcr = 2.8 MPa φc = 0.75

Prestressing tendon properties

fpy = 1670 MPa φp = 0.95 Number of strands = 136 Ap = area of prestressing tendons = (136 strands)(140 mm

2) = 0.019 m

Cross-sectional properties

b = width of compressive component = 15 m d = distance from extreme compression fibre to centroid = 2.19 m y = distance from base to centroid = 1.69 m Ig = moment of inertia = 6.06 m

4

Flexural strength requirements α1 = 0.85 − 0.0015f ′ c = 0.78 [CHBDC, Cl. 8.8.3] β1 = 0.97 − 0.0025f ′ c = 0.85 T = ϕp fpy Ap

= 0.95 1670 MPa 0.019m2 = 30 200 kN

a =T

ϕcα1f ′ cb

=30 200 kN

0.75 0.78 50 MPa 15 m

= 0.688 m

z = d −a

2

= 2.19 m −0.0688 m

2

= 2.16 m

c

εc= 3.5x10-3

φ a =β

1c

fpy

α1f

c’

T = φpA

pf

py

C

z Mr= Tz

Strain Stress Forces

120

Mr = T ∙ z = 30 200 kN 2.16 m = 65 100 kNm

Mr = 65 100 kNm > MULS = 53 000 kNm ∴ number of tendons is sufficient

Minimum reinforcement requirement

Mr > 𝑚𝑖𝑛 1.2 Mcr

1.33 MULS

[CHBDC, Cl.8.8.4.3]

1.2 Mcr =fcr Ig

y = 12 000 kNm

1.33 MULS = 70 500 kNm Mr = 53 000 kNm > 1.2 Mcr = 12 000 kNm ∴ requirement is satisfied

Maximum reinforcement requirement

c

d< 0.5 [Cl.8.8.4.5]

c =a

β1

=0.0688 m

0.85= 0.081m

c

d=

0.081 m

2.91 m= 0.037 < 0.5 ∴ 𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑚𝑒𝑛𝑡 𝑖𝑠 𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑑

B.2 Shear Strength at ULS

This calculation is performed for a cast-in-place on falsework box-girder analysis case with span length L of 50m and span-to-depth ratio L/h of 20. The following calculation aims to find the minimum stirrup spacing s needed to satisfy shear requirements.

VULS = shear demand = 11 300 kN MULS = moment demand = 5330 kNm Vp = component of prestressing force in the direction of shear = -3000 kN Mp = prestressing moment = 8300 kNm

Concrete properties

f’c = 50 MPa fcr = 2.8 MPa φc = 0.75 ag = aggregate size = 10 mm

Prestressing tendon properties

fpy = 1670 MPa φp = 0.95 Ep = 200 000 MPa Number of strands = 136 Ap = area of prestressing tendons = (136 strands)(140 mm

2) = 0.019 m

2

Aps = area of prestressing tendons in the flexural tension side = 0.568 m2

Non-prestressed reinforcement properties

fy = 400 MPa φs = 0.90

121

Es = 200 000 MPa Av = area of stirrup = 0.0012 m

2 (4-20M stirrups)

As = area of longitudinal non-prestressed reinforcement = 0.01 m2

Cross-sectional properties

b = width of compressive component = 15 m d = distance from extreme compression fibre to centroid = 1.31 m h = height of section = 2.5 m

dv = max 0.72 h0.9 d

= 1.8 m

bv = minimum web width within dv = 0.938 m sz = crack spacing parameter = dv = 1.8 m

sze =35sz

15 + ag = 2.52 m

fpo = 0.7fpu for bonded tendons

σp,∞ for unbonded tendons= 1300 MPa

εx =

Mf

dv+ Vf − Vp − Aps fp0

2 EsAs + EpAps = −0.003 = 0, 0 < εx < 3.0 × 10−3

θ = 29 + 7000εx 0.88 +sze

2500 = 55°

β = 0.4

1 + 1500εx

1300

1000 + sze = 0.148

Vc = concrete shear resistance = 2.5βϕcfcr bv dv = 3550 kN Vr = Vc + Vs + Vp

Vs = shear resistance provided by stirrups = Vr − Vc − Vp = 10 800 kN

Vs =∅sfy Av dv cotθ

s

Av

s=

Vs

∅s fy Av dv cotθ= 0.0238 m

s =Av

0.0238 m= 420 mm > 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑠𝑝𝑎𝑐𝑖𝑛𝑔 𝑜𝑓 300 𝑚𝑚 ∴ 𝑠 = 300 𝑚𝑚 𝑖𝑠 𝑢𝑠𝑒𝑑

122

B.3 Thermal Gradient Moments

This section describes the derivations of moments and stresses due to thermal gradients.

Final curvature of bending ψ:

ψ =ε y + ε0

y

The primary stresses: fr y = Ec ε y − εf

= Ec ε y − αcθ y = Ec ε0 + ψy − αcθ(y)

where αc = thermal coefficient of concrete = 10 × 10−6/℃ [CHBDC, Cl.8.4.1.3]

Axial restraint force Pr and restraint moment Mr:

Pr = fr y b y dyh

0


0

b y dy

Mr = fr y b y y − n dyh

0


0

b y y − n dy

Note the following relationships:

yb y dy = nA

y2b y dy = I + n2A

b y dy = A

Now, equate both Pr and Mr to zero because there are no external forces and internal redundancies have been removed to make the structure statically-determinate. First, Pr is equated to zero to obtain the final strain at elevation y=0 (ε0):

Pr = 0

fr y b y dyh

0

= 0

Ec ε y − αcθ y h

0

b y dy = 0

ε0 + ψy − αcθy b y dyh

0

= 0

ε0b y dy + ψyb y dyh

0

− αcθyb y dyh

0

h

0

= 0

ε0A + ψnA − αcθyb y dyh

0

= 0

123

ε0 =αc

A θ y b y dy

h

0

− nψ

Similarly, Mr is equated to zero to obtain the final curvature of bending ψ: Mr = 0

fr y b y y − n dyh

0

= 0

Ec ε y − αcθ y h

0

b y y − n dy = 0

ε0 + ψy − αcθ y h

0

b y y − n dy = 0

ε0yb y dyh

0

− ε0b y ndyh

0

− αcθ y b y ydyh

0

+ nαcθ y b y dyh

0

+ ψy2b y dy − ψyb y ndyh

0

h

0

= 0

ε0nA − ε0nA − αcθ y b y ydyh

0

+ nαcθ y b y dyh

0

+ ψ I + n2A − ψn2A = 0

nαcθ y b y dyh

0

− αcθ y b y ydyh

0

+ ψI = 0

ψ =αc

I θ y b y y − n dy

h

0

To remove this incompatible rotation, restraint moment M is needed at each end of the span:

M = −EcIψ

This restraint moment is distributed to obtain the final moments caused by thermal gradient Mfinal. This final moment causes secondary stresses fs:

fs y =Mfinal (y − n)

I

The total thermal stresses that need to be considered in design checks are computed with the following equation:

ftemp y = fr y + fs y

= Ec ε0 + ψ ∙ y − αcθ(y) +Mfinal (y − n)

I

124

B.4 External Tendon Force

This section describes an iterative process to calculate the prestressing force P in unbonded tendons to equilibrate external load Q at ULS.

This method requires the knowledge of prestressing steel area and effective prestress after all losses. First, the force in the unbonded tendons is assumed to be Pi. Pi must be less than the yielding force Py but greater than the minimum force Pmin needed to equilibrate the external load Q. Then the change in length due to force ∆lPF is computed using:

∆lPF =(Pi − P∞)

APEP

lP0

where P∞ = effective prestress force after all losses = σp,∞AP = 60%fpu AP lP0 = length of tendon when prestress force is P∞

The change in length due to deformation ∆lPD, which is the actual length of tendon when loaded by Q and Pi, is also computed:

∆lPD = εcP x dx

where εcP = concrete strain at PT level due to Q and Pi If the two changes in length are not equal, Pi is varied and the calculations are performed again. This process is iterated until ∆lPF equals ∆lPD, and Pi at this point is the actual force in the tendons at ULS. This iterative process is illustrated in the following figure.

Q

x

P P

Q Q

εc

εcp

= ∆εp

εcp

C

T=Apf

py when ∆ε

p+ε

p∞>ε

py

εcp

εp∞

εp,total

= ∆εp+ ε

p∞ = ε

cp+ε

p∞

Total tendon strain varies along the tendon length

εc

εcp

≠∆εp

C

T

εcp

εcp, average

=∆εp

εcp, average

= ∆εp= total deformation/undeformed length of tendon

εcp, average

=∆εp

εp∞

εp,total

=∆εp+ ε

p∞= ε

cp, average+ε

p∞

Total tendon strain remains constant along the tendon length.Tendon stress is averaged out between adjacent anchorages.Therefore, the flexural resistance is lower at critical locat ionscompared to the bonded case.

Pmin

Py

PULS

-20

0

20

40

60

80

100

120

40000 42000 44000 46000 48000 50000

Deformation(mm)

P (kN)

∆lPF

∆lPD

∆lPF

∆lPD

125

B.5 Total Construction Cost

The following calculation is performed for a cast-in-place on falsework box-girder with span length L of 35m and total bridge length of 260m. Material unit prices Concrete = $ 1500/m3 Prestressing tendons = $ 8.5/kg Reinforcing steel = $ 5.0/kg Other construction costs are more dependent on the span length and are assumed to be the same for all cases. They are computed based on the total superstructure cost of the baseline case. Item Unit L/h

10 15 20 (baseline) 25

Concrete ($ 1500/m

3)

Concrete volume m3 2700 2410 2260 2170

Total concrete cost $ 4,050,000 $ 3,610,000 $ 3,390,000 $ 3,260,000

Prestressing tendon ($ 8.5/kg)

Tendon length per strand m 359 359 358 358

Number of strands 52 76 108 136

Total tendon length m 18700 27300 38700 48700

Tendon mass kg 20600 30100 42600 53600

Total prestressing tendon cost $ 175,000 $ 255,000 $ 362,000 $ 456,000

Reinforcing steel ($ 5.0/kg)

Mass of reinforcing steel per concrete volume

kg/m3 80.0 80.0 82.0 86.1

Reinforcing steel mass kg 216000 193000 186000 187000

Total reinforcing steel cost $ 1,080,000 $ 963,000 $ 928,000 $ 936,000

Total superstructure cost (54.5% of total construction cost)

$ 5,310,000 $ 4,830,000 $ 4,680,000 $ 4,650,000

Other construction cost (e.g. substructure, mobilization, accessories) (45.5% of total construction cost)

$ 3,910,000 $ 3,910,000 $ 3,910,000 $ 3,910,000

Total construction cost $ 9,240,000 $ 8,740,000 $ 8,590,000 $ 8,560,000

Total construction cost per deck area per m2 $ 2,360 $ 2,240 $ 2,200 $ 2,200

126

APPENDIX C: SUMMARY OF RESULTS

1

27

C.1 Cast-in-Place on Falsework Box-Girder

Table C-1. Summary of results of cast-in-place on falsework box-girder analysis

Span-to-depth ratio

Span length (m)

35 50 60 75

10 $ 2370 (107%) 4 15-strand ducts

$ 2670 (109%) 4 19-strand ducts

$ 2890 (112%) 4 27-strand ducts

$ 3280 (111%) 4 37-strand ducts

15 $ 2250 (102%) 4 19-strand ducts

$ 2490 (102%) 4 27-strand ducts

$ 2670 (103%) 4 31-strand ducts

$ 3030 (103%) 8 27-strand ducts

20 $ 2210 (100%)

4 27-strand ducts

$ 2450 (100%) 8 19-strand ducts

$ 2590 (100%) 8 27-strand ducts

$ 2950 (100%) 8 37-strand ducts

25 $ 2200 (99.4%)

8 19-strand ducts

$ 2450 (100%) 8 27-strand ducts

$ 2580 (99.5%) 8 27-strand ducts

$ 2950 (100%) 12 27-strand ducts

30 $ 2490 (102%)

8 37-strand ducts

$ 2600 (100%) 8 37-strand ducts

$ 3020 (103%) 12 37-strand ducts

35 $ 2660 (103%) 12 27-strand ducts

$ 3220 (109%) 16 37-strand ducts

1

28

C.2 Cast-in-Place on Falsework Solid Slab

Table C-2. Summary of results of cast-in-place on falsework solid slab analysis

Span-to-depth ratio

Span length (m)

20 25 30 35

30 $ 2270 (100%)

10 27-strand ducts

$ 2650 (100%) 10 27-strand ducts

$ 3000 (100%) 12 27-strand ducts

$ 3450 (100%) 18 27-strand ducts

35 $ 2240 (98.1%)

12 27-strand ducts

$ 2560 (96.6%) 12 27-strand ducts

$ 2890 (96.3%) 14 27-strand ducts

$ 3320 (96.3%) 20 27-strand ducts

40 $ 2240 (98.8%)

14 27-strand ducts

$ 2520 (95.2%) 14 27-strand ducts

$ 2830 (94.2%) 18 27-strand ducts

$ 3250 (94.3%) 22 27-strand ducts

45

$ 2860 (95.4%) 20 27-strand ducts

$ 3280 (95.1%) 24 27-strand ducts

1

29

C.3 Precast Segmental Span-by-Span Box-Girder

Table C-3. Summary of results of precast segmental span-by-span box-girder analysis

Span-to-depth ratio

Span length (m)

30 40 50

15

$ 2040 (100%) 6 27-strand ducts

$ 2280 (100%) 6 37-strand ducts

$ 2380 (100%) 6 37-strand ducts

20

$ 2040 (100%) 6 37-strand ducts

$ 2250 (98.6%) 8 37-strand ducts

$ 2350 (98.4%) 8 37-strand ducts

25

$ 2280 (100%) 8 37-strand ducts

$ 2390 (100%) 9 37-strand ducts

130

C.4 Sensitivity with Respect to Changes in Construction Cost Breakdown

Figure C-1. Cast-in-place on falsework box-girder with L=50m

Figure C-2. Cast-in-place on falsework solid slab with L=25m

0.3% (6.9%)40%

0.3% (8.6%)50%0.4% (9.0%)54.5% (Menn)0.4% (10%)60%0.5% (12%)70%0.5% (14%)80%

0.1% (3.4%)

Cost variation Superstructure as % of total construction cost

20%

0.2% (5.2%)30%

$0

$2,000

$4,000

$6,000

$8,000

0 5 10 15 20 25 30 35


($/m2)

L/h

3.4% (3.5%) 40%

4.3% (4.5%) 50%4.8% (5.0%) 54.5% (Menn)5.1% (5.4%) 60%6.0% (6.4%) 70%6.8% (7.3%)

80%

1.7% (1.7%)

Cost variationSuperstructure as % of total construction cost

2.6% (2.6%)

20%

30%

$0

$2,000

$4,000

$6,000

$8,000

25 30 35 40 45


($/m2)

L/h

131

13

1

Figure C-3. Precast segmental span-by-span box-girder with L=40m

0.6% (1.7%) 40%

0.8% (2.1%) 50%0.8% (2.3%) 54.5% (Menn)0.9% (2.5%) 60%1.1% (3.0%) 70%1.2% (3.4%) 80%

0.3% (0.8%)

Cost variationSuperstructure as % of total construction cost

0.5% (1.3%)

20%

30%

$-

$2,000

$4,000

$6,000

$8,000

10 15 20 25 30


($/m2)

L/h

Poon Sandy SY 200911 MASc Thesis (1)

Documents